








\0 



-x 



v. 






**. 
























^ ^ 






^ v* 



^ ^ 






















a 0&> 










> V 




'^a 




















»1H* ^ s ., 


■^ '. 


















.<5 <> 




.o> «c„ 



, ■ 



,0(5 






<£\ 



^ ^ 



.0o, 









,0(5 



A 






oH -7\ 









^' 





















\° o* 









<s V 






oS -Kt 









o o x 



«S "^ 









.V 



\V 















V 



V- v 






x o 






^ v* 






<"£. .A 






-Pi. y 



■^ 



</> \\ 




















"c 















^ V? 









.0 o. 






<■ 



'** *& 






** „** 



,0o. 



^ ^ 



V ^ - 



\\^ 









*«K 



^ v 



v><s. 



^ ^ 















,0 e> 



%• "£" 









' 






■^ o X 






/? 



PHYSICAL 



AND 



CELESTIAL MECHANICS. 



BY 

BENJAMIN PEIRCE, 

PERKINS PROFESSOR OF ASTRONOMY AND MATHEMATICS IN HARVARD UNIVERSITY, 

AND CONSULTING ASTRONOMER OF THE AMERICAN EPHEMERIS 

AND NAUTICAL ALMANAC. 



\H 



DEVELOPED IN FOUR SYSTEMS OF 



ANALYTIC MECHANICS, CELESTIAL MECHANICS, POTENTIAL 
PHYSICS, AND ANALYTIC MORPHOLOGY. 



BOSTON: 
LITTLE, BROWN AND COMPANY. 

1855. 



A SYSTEM 



ANALYTIC MECHANICS. 



BY 



BENJAMIN PEIRCE 



PERKINS PROFESSOR OF ASTRONOMY AND MATHEMATICS IN HARVARD UNIVERSITY, 

AND CONSULTING ASTRONOMER OF THE AMERICAN EPHEMERIS 

AND NAUTICAL ALMANAC. 



BOSTON: 
LITTLE, BROWN AND COMPANY. 

1855. 



Entered according to Act of Congress in the year 1855, by 

LITTLE, BROWN AND COMPANY, 

In the Clerk's Office of the District Court of the District of Massachusetts. 



CAMBRIDGE: 

ALLEN AND FARNHAM, PRINTERS. 



TO 



THE CHERISHED AND REVERED MEMORY OF 



MY MASTER IN SCIENCE, 



NATHANIEL BOW DITCH, 



THE FATHER OF AMERICAN GEOMETRY, 



THIS VOLUME 



IS INSCRIBED. 



ADVERTISEMENT. 



The substance of the present volume was originally pre- 
pared as part of a course of lectures for the students of mathe- 
matics in Harvard College. But at the request of some of my 
pupils, and especially of my friend Mr. J. D. Runkle, I have been 
induced to undertake its publication. The liberality of my 
publishers, the well-known firm of Little, Brown & Co., who gen- 
erously gave directions to the printers, that no expense should be 
spared in its typographical execution, seemed to impose upon me 
an increased obligation to perform my portion of the task to 
the best of my ability. I have consequently reexamined the 
memoirs of the great geometers, and have striven to consoli- 
date their latest researches and their most exalted forms of 
thought into a consistent and uniform treatise. If I have, 
hereby, succeeded in opening to the students of my country a 
readier access to these choice jewels of intellect, if their bril- 



viii ADVERTISEMENT. 

liancy is not impaired in this attempt to reset them, if in 
their new constellation they illustrate each other and concen- 
trate a stronger light upon the names of their discoverers, and 
still more, if any gem which I may have presumed to, add, is 
not wholly lustreless in the collection, I shall feel that my 
work has not been in vain. The treatise is not, however, 
designed to be a mere compilation. The attempt has been 
made to carry back the fundamental principles of the science 
to a more profound and central origin ; and thence to shorten 
the path to the most fruitful forms of research. It has, 
moreover, been my chief object to develop the special forms 
of analysis, which are usually neglected, because they are only 
applicable to particular problems, and to restore them to their 
true place in the front ranks of scientific progress. The 
methods which, on account of their apparent generality, have 
usually attracted the almost exclusive attention of the student, 
are, on the contrary, reestablished in their true position as 
higher forms of speciality. 

BENJAMIN PEIRCE. 



LIST OF SUBSCRIBERS. 



J. I. Bowditch, (10 copies), 


Boston. 


John D. Runkle, (5 copies), 


Cambridge. 


Chauncey Wright, (2 copies), 


a 


C. H. Sprague, (2 copies), 


Maiden. 


W. C. Kerr, 


Davidson College 


George Eastwood, 


Saxonville. 


Charles Phillips, 


Chapel Hill, N. C, 


Joseph W. Sprague, (2 copies), 


Rochester, N. Y. 


J. M. Chase, 


Cambridge. 


B. H. Chase, 


u 


Sharon Tynclale, 


a 


Isaac Bradford, 


a 


John Bartlett, (3 copies), 


u 


Gustavus Hay, 


Boston. 


F. J. Child, 


Cambridge. 


William C. Bond, for Observatory of 




Harvard College, (2 copies), 


u ■ 


J. E. Oliver, (2 copies), 


Lynn. 


C. W. Little, 


Cambridge. 


N. Hooper, 


Boston. 


C. F. Choate, (2 copies), 


Cambridge. 



X 



LIST OF SUBSCRIBERS. 



J. P. Cooke, Jr., 


Cambridge. 


B. A. Gould, Jr., 


u 


Joseph Winlock, 


(C 


H. L. Eustis, 


u 


Joseph Lovering, 


u 


C. Gordon, 


a 


Jared Sparks, 


u 


A. Brown, 


a 


William G. Choate, 


a 


J. F. Flagg, Jr., 


Washington, D. C. 


A. E. Agassiz, 


Cambridge. 


William G. Pearson, 


« 


Charles Sanders, (2 copies), 


« 


Theophilus Parsons, 


a 


John Erving, Jr., 


a 



Charles H. Mills, 

Edmund D wight, 

Edward Everett, 

J. H. C. Coffin, 

T. S. Hubbard, 

Morclecai Yarnall, 

James Major, 

James Ferguson, 

B. B. Hamilton, 

Stephen Alexander, (2 copies), 

James Walker, 

William Chauvenet, (4 copies), 

Washington Observatory, (5 copies), 

Thomas Hill, 

Waltham Rumford Institute, 

Charles Avery, 



Boston. 



Annapolis, Md. 
Washington, D. C. 



Syracuse, N. Y. 
Princeton, N. J. 
Cambridge. 
Annapolis, Md. 
Washington, D. C. 
Waltham. 



Hamilton College, N. Y. 



LIST OF SUBSCRIBERS. 



XI 



John Paterson, 

Albany State Library, 

State Normal School, 

George R. Perkins, 

A. D. Bache, (4 copies), 

American Nautical Almanac, (5 copies), 

J. W. Jackson, 

Robert S. iVvery, 

A. W. Smith, 

J. M. Vanvleck, (2 copies), 

Sarah Watson, 

Smithsonian Institution, (25 copies), 

George W. Phillips, 

H. Bailliere, (3 copies), 

E. S. Snell, 

John F. Frazer, 

N. Fisher Longstreth, 

Fairman Rogers, 

E. Otis Kendall, 
Ira Young, 

P. H. Sears, 

W. F. Phelps, 

Edward L. Force, 

Didactic Society of University of N. C., 

F. B. Downes, 
W. G. Peck, 
Elias Loomis, 
John Tatlock, 
Edward Pearce, 
James Mills Peirce, 
C. W. Eliot, 



Albany, N. Y. 



Utica, N. Y. 
Washington, D. C. 
Cambridge. 
Union College, N. Y. 
Washington, D. C. 
Middletown, Conn. 



Nantucket. 

Washington, D. C. 

Salem. 

New York City. 

Amherst. 

Philadelphia. 



Dartmouth College. 
Boston. 
Albany. 

Washington, D. C. 
Chapel Hill, N. C. 
Racine, Wis. 
West Point, N. Y. 
New York City. 
Williamstown College. 
Providence, R. I. 
Cambridge. 



XII 



LIST OF SUBSCRIBERS. 



F. W. Bardwell, 


Cambridge. 


Charles L. Fisher, 


« 


William Dearborn, 


Boston. 


Sidney Coolidge, 


« 


L. E. Gibbs, 


Charleston, S. C. 


J. D. Crehore, 


Newton. 


J. G. Cogswell, Astor Library, 


New York City. 


Wolcott Gibbs, 


a a 


Trubner & Co., (25 copies), 


London. 


William S. Haines, 


Providence, R. I. 


Alexis Caswell, 


a u 


George M. Hunt, 


Stuyvesant, N. Y. 


Charles C. Snow, 


Brooklyn, N. Y. 


B. Westermann, 


New York City. 


W. H. Cathcot, 


Urbanna University, Ohio, 


S. C. Huntington, 


Pulaski, Orange Co., N. Y. 


Edward King, 


New York City. 


A. G. Harlow, 


Cambridge. 


C. A. Cutter, 


u 


R T. Paine, 


a 


C. F. Sanger, 


a 


John B. Tileston, 


a 


J. M. Sewall, 


a 


B. S. Lyman, 


a 


C. H. Davies, 


u 


Isaac A. Hagar, 


u 


F. H. Smith, 


University of Virginia. 



ANALYTICAL TABLE OF CONTENTS. 



CHAPTER I. 

MOTION, FORCE, AND MATTER. 



Section Page 

1. The universality of motion, . . . 1 

2. The spiritual origin of force, . . 1 



Section 
3. The inertia of matter, . 



Page 
. 1 



CHAPTER II. 

MEASURE OF MOTION AND FORCE. 



I. The Measure of Motion. 

4. Uniform motion, .... 2 

5. Velocity defined, . . . .2 

6. Formula of varying velocity (2 27 ), . 2 

II. The Measure of Force. 

7. Force and power, .... 2-3 

8. The force is proportional to the ve- 

locity, . . . . . .3 



9. The mass defined, . 
40. The inertia of a body, 



3 

3-4 



III. Force of moving Bodies. 

11. The power is proportional to the 

square of the velocity, ... 4 

12. The power is half the product of the 

force by the velocity, ... 4 

13. Formula of varying force, . . 5 



CHAPTER III. 

FUNDAMENTAL PRINCIPLES OF REST AND MOTION. 



I. Tendency to Motion. 

14. The total force of a system of bodies 

is the exact equivalent of the sum 

of its components, .... 5 

15. The fundamental principle of equi- 

librium, ..... 5 

16. The measure of the tendency to any 

form of motion, .... 5-6 

17. Formula for the measure of the ten- 



dency to a proposed motion, ex- 
pressed by virtual velocities (7 12 ), 6-7 

II. The Equations of Motion and Best. 

18. The equation of motion (8 18 ) and that 

of rest (8o ), 7-8 

19. These equations can be decomposed 

into as many partial equations as 
there are independent elements, . 8-9 



XIV 



ANALYTICAL TABLE OF CONTENTS. 



CHAPTER IV. 

ELEMENTS OF MOTION. 



20. 



21. 



22. 



I. Motion of Translation. 

A point has three independent ele- 
ments of motion, from which any 
other elementary motion can be 
obtained by the formula (10 23 ), 9-10 

Translation and rotation, being almost 
universal, are the most important 
forms of motion, . . . 10-11 

Definition of translation, . . .11 



23. Parallelopiped of translation, 



11 



II. Motion of Rotation. 

24. Definition of rotation, . . .12 

25. Projection of rotation upon any axis, 12-13 
26', 27. Parallelopiped and parallelogram 

of rotations, .... 13-14 
28, 29. Elementary rotations projected 

upon a given direction, . . 14-15 
30, 31. Theorem of two systems of rectan- 
gular axes (15 28 ), . . . 15-16 

III. Combined Motions of Rotation and 

Translation. 

32. Decomposition of a rotation into a ro- 

tation about a parallel axis and 
a translation in a perpendicular 
plane, 16 

33. Combination of rotations about par- 

allel axes, . . . . .16 

34. Equations of the axis of combined 

parallel rotations, . . . 16-17 

35. Combined equal rotations about par- 

allel axes, . . . . .17 

36. Combined rotations about opposite 

axes, 17-18 

37. Combined rotations which are equiva- 

lent to a translation, . . .18 



A couple of rotations, 

Combination of rotation and transla- 
tion, ...... 

Analysis of every possible motion of 
a solid, into a screw motion, . 18 

The instantaneous axis of rotation, . 

Special mode of conceiving the motion 
of a solid by means of the surfaces 
described by the instantaneous 
axis, 

Case in which the surfaces of § 42 are 
developable, . . . .19 

Case in which the surfaces of § 42 are 
cylinders or cones, . . . 20-21 

General case reduced to that of §44, 
combined with translation, 

Axes of greatest curvature of the coni- 
cal surface, .... 

Decomposition of the rotation into 
rotations about the axes of greatest 
curvature, 

Relations of the velocities of rotation 
to that of the instantaneous axis, 22-23 

Relations of the rotations when the 
surfaces are cylinders, . . .23 



19 



20 



21 



21 



22 



IV. 



Special Elements of Motion and 
Equations of Condition. 



50. The independent elements of position, 24 

51. The equation of condition for depend- 

ent elements, ... . . 24-25 

52. 53. Elimination by the method of mul- 

tipliers, 25-26 

54. The variation of the equation of con- 
dition expressed by means of the 
variation of the normal to a cor- 
responding surface, . . .26-27 



ANALYTICAL TABLE OF CONTENTS. 



XV 



CHAPTER V. 

FORCES OF NATURE. 



I. The Potential, Level Surfaces, Posi- 
tions of Equilibrium, and the Pos- 
sibility of Perpetual Motion. 

55. The fixed laws are not incompatible 

with the spiritual origin of force, . 28 

56. Fixed and variable forces, . . 28 
5 7. The relation of the forces of nature to 

form expressed by the potential, 28-29 

58. The dependence of the power of a 

system upon its form, . . .29 

59. Limits of motion of a system, . . 29 
GO, 61. The potential is a maximum or a 

minimum for the position of equi- 
librium, 29-30 

62. The relation of stability of equilibrium 

to the maximum or minimum of 
the potential, .... 

63. There are as many positions of stable 

as of unstable equilibrium, witli ref- 
erence to each element of equi- 
librium, 

64. The necessity of the potential in the 

fixed forces of nature, and its rela- 
tion to the possibility of perpetual 
motion, ...... 

65. The level surface and its finite extent, 

66. The direction of attraction is perpen- 

dicular to the level surface, . 

67. The law of attraction determined by 

the distance apart of two infinitely 
near level surfaces. Level surfaces 
do not intersect each other. The 
continuity of the potential of nature, 

68. The trajectory of level surfaces termi- 

nates in a maximum or minimum, 32-33 

69. The limits in space of the constant 

potential coincide with those of the 
discontinuity of the potential or its 
derivatives, ..... 

70. There is no force or mass throughout 

a space of constant potential, 

71. The potential of nature and its deriva- 



30 



30 



32 



33 



33 



tives are finite and continuous 
throughout a space which contains 
no mass, . . . . .33 

72. A portion of space, for which the po- 

tential of the fixed forces of nature 
is constant, is completely bounded 
by a continuous mass, . . 33-34 

73. The potential of nature for tempora- 

rily fixed forces may vanish for an 
infinite extent of space, . . 34 

74. 75. The computation of the difference 

of the potential for two points by 
the formula (34™,), ... 34 

II. Composition and Resolution of 
Forces. 



35 



35 



76. All the phenomena of nature depend 

upon combined forces, . 

77. The projection of a force in a given 

direction, ..... 

78. The action of a combination of forces 

in any direction, . . . 35-36 

79. The parallelopiped offerees, . . 36 

80. 81. The resultant of forces, and its al- 

gebraic expression (37 25 ), . . 36-37 

82. The tendency of a system to a motion 

of translation, .... 37-38 

83. The moment of a force, . . .38 

84. The projection of a moment, . . 38-39 

85. The parallelopiped of moments, . . 39 

86. The moment of a force measures its 

tendency to produce rotation, 

87. The positive direction of the axis of a 

moment, ..... 

88. The resultant moment measures the 

total tendency to produce rotation, 

89. The resultant moment of forces which 

act upon a point is the moment of 
their resultant, .... 

90. 91. The moments for parallel lines, 
92. The resultant moment for different 

points, ...... 



39 



39 



39 



40 
40 

40 



XVI 



ANALYTICAL TABLE OF CONTENTS. 



93. A couple of forces, ... 40 

94. The moment of a couple is constant 

for all points of space, . . .40 

95. The tendency of a system of forces to 

produce translation and rotation 
may be reduced to a resultant and 
a resultant couple, . . . 40-41 

96. It may be still further reduced to 

two forces, . . . . .41 

97. The resultant and the resultant 

moment may always coincide in 
direction, 41 

98. 99. If the forces are in the same plane, 

or if they are parallel, the combined 
equivalent is either a resultant or a 
resultant moment, . . .42 

100. Analytic determination of the com- 

mon direction of the resultant and 
resultant moment (43 ), . . 42-43 

101. The special reduction of forces re- 

quires special forms of analysis, . 43 

III. Gravitation, and the Force of 
Statical Electricity. 

102. Gravitation, and its elementary po- 

tential, . . . . . .43 

103. Statical electricity, and its element- 

ary potential, .... 43-44 

104. Law of distribution of electricity, . 44 

105. Potential of gravitation and electric- 

ity (45,), 44-45 

10G. Laplace's equation for the determi- 
nation of the potential (46 5 ), . 45-46 

107. The law of attraction of gravitation 

or electricity (46 17 ), . . .46 

108-112. The Attraction of an Infi- 
nite Lamina, .... 46-48 

108. The potential of an infinite lamina, 46-47 

109. The level surfaces of a uniform lamina, 47 
110-112. The attraction of a uniform la- 
mina (48 4 ) and (48 lc ), . . 47-48 

113. Poisson's Modification of La- 
place's Equation for an In- 
terior Point (49.,), . . 48-49 

114-124. The Attraction of an In- 
finite Cylinder, . . . 49-54 



114. The form of the potential of an in- 

finite cylinder (50 31 ), . . . 49-50 

115. The level surfaces of an infinite cyl- 

inder, 50 

116. Form of the attraction of an infinite 

cylinder (50 7 _ li; ), . . . .50 
117-120. The attraction of an infinite cyl- 
inder upon a distant point (51*,), 

(52 n ), 50-52 

121-123. The attraction of a circular cyl- 
inder (53 30 ), (54,), (54 7 ), (54 16 ), 52-54 

125, 126. Relation of the Poten- 
tial to its Parameter, . 54-55 

127, 128. Attraction of a Finite 
Point upon a Distant Mass. 
The Centre of Gravity, . 55-56 

129-132. The Attraction of a Spher- 
ical Shell (56 31 ), (57 19 ), (57 M ), 
(58 8 ), 56-58 

133-147. The Action and Reaction 
of a Surface or infinitely 
thin Shell of finite extent. 
The Chaslesian Shell, . 58-69 

133. The total action of a surface normal 

to itself, 58-59 

134. That of a plane, . . . .59 

135. Gauss's theorem relative to the angle 

subtended by a surface (60 24 ), . 59-60 

136. Gauss's and Chasles's theorem up- 

on the normal action of masses up- 
on a surface (61 15 ), . . . 60-61 

137. Any level surface must inclose masses 

of matter, ..... 61-62 

138. The potential of a closed surface 
, which is level to itself is constant 

for the inclosed space, . . .62 

139. The maximum limit of the potential 

is within the mass, . . . .62 

140. In a gravitating system there is no 

point of minimum potential, . . 62 

141. The attraction is constant upon all 

the sections of a trajectory canal 
by a level surface, . . . 62-63 

142. Condensed view of the laws of attrac- 

tion, and their relation to the pro- 
pagation of heat, . • ■ 63-64 



ANALYTICAL TABLE OF CONTENTS. 



XV11 



143. Correspondence of the Chaslesian 

shells upon different level sur- 
faces, . . . \ . . . 64-65 

144. The ratio of the mass of a Chaslesian 

shell to the inclosed mass, . .65 

145. The law of attraction of a Chaslesian 

shell (67 9 ), (6 7, 3 ), . . . 65-68 

146. The surface which is level to itself is 

a Chaslesian shell, . . .68 

147. Chaslesian shells upon the same sur- 

face are similar, . . . .69 

148-177. The Attraction of an 

Ellipsoid, .... 69-88 

148. The Chaslesian ellipsoidal shell, 69-70 

149. The Newtonian shell, ... 70 

150. Ivory's corresponding points of 

ellipsoids, 70-71 

151. The corresponding elements of New- 

tonian shells are proportional to 
the shells, 71-72 

152. Homofocal Newtonian shells, . . 72 

153. Corresponding projections of corre- 

sponding radii vectores of Newton- 
ian shells, 72-73 

154. Difference of the squares of corre- 

sponding radii vectores, . . 73 

155. Distance of corresponding points, . 73-74 

156. The external level surfaces of ellip- 

soidal Chaslesian shells, . . 74 

157. Tlie attractions of homofocal Newto- 

nian shells, ..... 74-75 

158. The attraction of a Chaslesian shell 

upon a point of its surface (7603), 75-76 

159. The attraction of a Chaslesian shell 

upon an external point (76 2 ), . 76-77 

160. There are three surfaces of the second 

degree which pass through a given 
point, and have given foci, of which 
one is ellipsoidal, and the other two 
are hyperboloids of different classes, 7 7 

161. The common intersection of the two 

hvperboloids cuts all homofocal 
ellipsoids in corresponding points, 77-78 

162. The three surfaces of § 60 cut each 

other rectangularly, . . . 78-79 

163. The common intersection of the two 

hvperboloids is a transversal to the 



level ellipsoidal surfaces of the 
same foci, . . . . .79 

164. Dupin's theorem that orthogonal sur- 

faces cut each other in the lines of 
greatest and least curvature, . 79-80 

165. Hyperboloidal Chaslesian shells, . 80 

166. 167. The attraction of an ellipsoid 

on an external point (82 24 ), (83 2 ), 80-83 

168. Legendre's formula for this attrac- 

tion (83 12 ), 83 

169. The expression of the attraction by 

elliptic integrals (84 18 ), (85 18 _ 24 ), 83-85 

170. Analytical theorems with reference 

to the attractions (86s_ 17 ), . . 85-86 

171. The attractions expressed as deriva- 

tives of a single function (86 25 _ 28 ), . 86 

172. The equation for limits of integra- 

tion (87 ), 87 

1 73. The condition that the attracted" point 

is upon the surface of the ellipsoid, 87 

174. The case in which the attracted 

point is within the ellipsoid, . .87 

175. The attraction when the density of 

the ellipsoid varies so that the com- 
ponent Chaslesian shells are homo- 
geneous, 87 

176. The attraction of a homogeneous ob- 

late ellipsoid of revolution, . 87-88 

177. The attraction of a homogeneous pro- 

late ellipsoid of revolution, . . 88 

178-218. The Attraction of a Sphe- 
roid. Legendre's and La- 
place's Functions, . . 88-116 

178. Jacobi's method adopted in the in- 

vestigation of the Legendre and 
Laplace functions, . . .88 
1 79-183. Investigation of the fundamental 
equations (89 18 ), (90 3 ), (90^,(90^), 
for the determination of the ele- 
ments of these functions, . . 89-90 

184, 186, 187. The relation of the suc- 

cessive elementary coefficients, and 
their derivatives (9 loo), (92 13 -93 24 ), 

91-93 

185. The Eulerian Gamma integral. See 

note, page 356, .... 91-92 
188, 189. The elementary functions of 



XV111 



ANALYTICAL TABLE OF CONTENTS. 



Legendre which vanish (93 31 ), 
(94 ), 93-94 

190. The general values of the elementary- 

functions for positive powers (942->), 94 

191. The elementary functions for the 

power of negative unity (95 20 ), . 94-95 

192. The elementary functions for nega- 

tive powers (97 4 ), . . . 95-97 

193. Development of the distance of two 

points according to the powers of 
their radii vectores, . . . 97-98 

194. The values of Legejstdre's or La- 

place's functions in this develop- 
ment (99 3 ), .... 98-99 

195. 201. Development of the potential 

of the spheroid, . . 99,102-103 

196. General form of these functions (100 6 ), 100 

197. Poisson's theorem for these func- 

tions, ..... 100-101 

198. General theorem for development by 

these functions, . . . .101 

199. 200. 'Laplace's theorems upon these 

functions, .... 101-102 

202. Development of the potential of the 

spheroid for an external point 
when the origin is the centre of 
gravity, 103 

203. The homogeneous ellipsoid which 

coincides in the first two terms of 
the development, . . . 103-107 

204. Development of the potential for an 

external point which is external to 
the spheroid as well as to the cor- 
responding homogeneous ellipsoid, 107 

205. The determination of the axes of the 

corresponding homogeneous ellip- 
soid, 107-108 

206-209. Determination of the potential 
. for a point which is quite close to 
the spheroid in Poisson's form of 
analysis, .... 108-112 

210. The attraction of a spheroid in the 

direction of the radius vector, 

211. The potential of the homogeneous 

spheroid for an external point, 

112- 

212. The potential of the homogeneous 

spheroid for a point of its surface, 113 



112 



-113 



213. Potential of the spheroid which dif- 

fers little from the ellipsoid, . .113 

214, 215, 216. Potential of a spheroid 

which is nearly a sphere, . 113-115 

217. Potential of a spheroid for an in- 

terior point, . . . . .116 

218. The discussion of the convergence 

of the series referred to subsequent 
volumes, . . . . .116 

IV. Elasticity. 

219. Nature of the phenomena of elastic- 

ity, 116-117 

220. Linear expansion of a body, and 

ellipsoid of expansion, . . 117-118 

221. Principal axes of expansion, . .118 

222. Surface of distorted expansion, 118-119 

223. Surface of distorted expansion re- 

ferred to principal axes, . .119 

224. 225, 226. Rotative effects of expan- 

sion, 119-120 

227. Total expansion of a body, . . 120 

228. Linear expansion for small disturb- 

ance, 120-121 

229. Reciprocal expansive ellipsoid for 

small disturbance, . . . .121 

230. Reciprocal expansive ellipsoid re- 

ferred to principal axes, . .121 

231. Case in which the reciprocal expan- 

sive ellipsoid becomes hyperboloid- 

al or cylindrical, . . . 121-122 

232. Total expansion for small disturb- 

ance, 122 

233. Rotation for small disturbance, . 122 

234. Directions of maximum, minimum, 

and mean rotation, . . 122-123 

235. Combination of mean rotations, . 123 

236. Rotation for small disturbance re- 

feiTed to principal axes, . .123 

237. Compression without mean rotation, 123 

238. Rotation without compression, . 123-124 

239. The discussion of the elastic force 

reserved for special chapter, . .124 

V. Modifying Fokces. 

240. Modifying forces defined, and di- 

vided into stationary and mov- 
ing, 124 



ANALYTICAL TABLE OF CONTENTS. 



XIX 



241. Eelation of stationary modifying 
forces to equations of condition, 

124-125 



242. Mode of action of modifying force, 

125-126 

243. Moving modifying forces, . .126 



CHAPTER VI. 



EQUILIBRIUM OP TRANSLATION. 



244. The conditions of equilibrium of 

translation (127 u ), . . .127 

245. The conditions of equilibrium of 

translation are the same as if all 
the forces were applied at a single 
point, 127-128 

246. Conditions of equilibrium of trans- 



lation when there is a fixed sur- 
face, line, or point, . . . 128 

247. The equilibrium of a material point 

wholly included in translation, . 128 

248. In the equilibrium of translation each 

force is equal and opposite to the 
resultant of all the others, . 128-129 



CHAPTER VII. 

EQUILIBRIUM OF ROTATION. 

249. The conditions of equilibrium of 



rotation, 129 

250. When the equilibrium of rotation is 

universal that of translation is in- 
volved, 129-130 

251. Equilibrium of rotation about parallel 

axes, . . . . . .130 

252. Axis for which the resultant moment 

vanishes, 130 



253. Equilibrium of rotation when there 

are two fixed points, . . . 130 

254. Relation of the centre of gravity to 

equilibrium of rotation of parallel 
forces, 130-131 

255. Internal forces neglected in the con- 

ditions of equilibrium of transla- 
tion or rotation, or action and re- 
action, are equal, . . . 131-132 



CHAPTER VIII. 

EQUILIBRIUM OP EQUAL AND PARALLEL FORCES. 



I. Maxima and Minima of the Potential. 

256. Gravitation taken as the type of these 

forces. The level surfaces are hor- 
izontal planes, . . . .132 

257. Relation of the maximum and mini- 

mum potential to equilibrium, 132-133 

258. The equilibrium of translation of 

gravitation requires stationary mod- 
ifying forces, 133 



259. The resultant moment of gravity 

vanishes for the centre of grav- 
ity, 133 

260. Position and magnitude of a single 

modifying force in a gravitating 
system, 133 

261. Position and magnitude of two mod- 

ifying forces in a gravitating sys- 
tem, 133 



XX 



ANALYTICAL TABLE OF CONTENTS. 



262. The change of the intensity of grav- 
ity does not affect the position of 
equilibrium, 



134 



II. The Funicular and the Catenary. 

263. The funicular defined, . . .134 

264. The funicular with one fixed point, 

134-135 

265. The funicular with two fixed points, 135 

266. The point of meeting of the lines of 

extreme tension of any portion of 
the funicular, . . . . 135-136 

267. The vertical projections of the ex- 

treme tensions with reference to 
the distance from the centre of 
gravity, . . . . . .136 

268. The inclination of the funicular to 

the horizon, .... 136-137 

269. Point of change in the funicular 

from ascent to descent, . . 137-138 

270. The equation of the funicular (138 14 ), 138 

271. The general equation of the cate- 

nary (138 31 ), 138 

272. The catenary of uniform chord (1 3 9 3 ), 139 

273. The uniform chord referred to rec- 

tangular coordinates (139 20 _ 24 ), . 139 

274. The tension of the uniform chord 

(139 31 ), 139 

275. The tension and thickness of a cate- 

nary of given form (140 4 _ 10 ), . . 140 

276. The catenary of uniform strength 

(140 20 _ 2r ), . . . . _ . 140 

277. The catenary for uniform support of 

weight (141^), . . . 140-141 

278. The elastic catenary (141,o_ 19 ), . 141 

279. The catenary upon a given surface 

(142 3 _ 7 ), 141-142 

280. The pressure of a catenary upon a 

surface (1422,,), . . . .142 

281. The point at which the curvature of 

the catenary upon a surface van- 
ishes (143 7 ), .... 142-143 

282. The catenary upon a vertical cylin- 

der (143 13 ), 143 

283. The catenary upon a vertical surface 

of revolution (143 19 ), . . .143 



144 



-146 



284. The case of a horizontal catenary 

upon a surface of revolution (143 28 ), 

143-144 

285. Direction of the catenary upon a sur- 

face of revolution (144^,), 

286. The catenary upon the vertical right 

cone with a circular base ; its equa- 
tion (145 12 ), and analysis into dis- 
tinct portions, . . .144- 

287. The finite portion of the catenary 

upon the vertical right cone ex- 
pressed by elliptic integrals (14 7 ), 

146-147 

288. The general expression of the arc of 

the spherical ellipse by elliptic in- 
tegrals (149 26 ), . . . 147-149 

289. The expression of the catenary upon 

the vertical right cone by the arc 

of the spherical ellipse (150 2 ), 149-150 

Cases in which the catenary upon 
the vertical right cone returns into 
itself, 150 

The infinite portions of the catenary 
upon the vertical right cone ex- 
pressed by elliptic integrals (150 31 ), 
(151 4 ), 150-151 

The finite and infinite portions of the 
catenary may be expressed by the 
aid of reciprocal spherical ellipses, 
(151 17 , 21 ), . . . . _. 

Case in which the finite portion is 
circular (151 29 ^i), .... 

Case in which the catenary upon the 
vertical cone degenerates into a 
straight line, ..... 

The investigation of the infinite por- 
tion of the catenary upon the verti- 
cal right cone, when the finite por- 
tion disappears (152 31 ), (153 7 ), 152-153 

Case, recognized by Bobillier, in 
which the catenary upon the verti- 
cal right cone becomes an equi- 
lateral hyperbola upon the devel- 
oped cone (15335), 

The catenary upon a vertical ellip- 
soid of revolution, its equation 

(15420), 

298. The cases in which there are two 



290. 



291. 



292. 



293. 



294. 



295. 



296. 



297. 



151 



151 



152 



153 



154 



ANALYTICAL TABLE OF CONTENTS. 



XXI 



portions of the catenary upon the 
vertical ellipsoid of revolution, or 
one portion, or when there is no 
catenary, .... 154-155 

299. The cases in which the two portions 

of the catenary upon the vertical 
ellipsoid of revolution are similar 
(156 35 ), (157 8i28 ). This case was 
recognized by Bobillier for the 
sphere, 155-157 

300. Expression of the constants by means 

of the limiting values in the general 
expression of the catenary upon 
the vertical ellipsoid of revolution, 

157-159 

301. Integral of the differential equation 



of the catenary in the general case 
of the vertical surface of revolu- 
tion (1592s), 159 

302. The catenary upon the vertical 

equilateral asymptotic hyperboloid 
when the inclination to the merid- 
ian is constant, . . . 159-160 

303. The definition of the catenary upon 

any vertical surface of revolution 
by means of the equilateral asymp- 
totic hyperboloid, .... 160 

304. The general case of the catenary 

upon the equilateral or asymptotic 
hyperboloid (161 12 ), . . 160-161 

305. The limiting point at which the cate- 

nary tends to leave the surface, . 161 



CHAPTER IX. 

ACTION OP MOVING BODIES. 



306, 307. Characteristic Function, 

162-163 

306. Maupertius's action of the system 

and Hamilton's characteristic 
functions (162 21 ), . . . 162 

307. Expenditure of action by a moving 

system (163 2 ), . . . 162-163 

308. Principle of Living Forces or 

Law of Power, (163 14 ) . 163 

309-312. Canonical Forms of the 
differential equations of 
Motion, .... 163-166 

309. Lagrange's canonical forms of 

equations of motion (164 12 ), 163-164 

310. Equations of motion expressed in rec- 

tangular coordinates (164 25 ), 164, 165 
311, 312. Hamilton's modifications of 'La- 
grange's canonical forms (165 37 ), 
(166 3 ), 165-166 

313-315. Variations of the Char- 
acteristic Function, 166-167 
313. Derivatives of the characteristic func- 



tion with reference to the elements of 
motion, .... 166-167 

314. Derivatives of the characteristic 

function for the rectangular ele- 
ments of motion, . . . 167 

315. Hamilton's method not applicable 

when the forces involve the veloc- 
ity, 167 

316-318. Principle of Least Action, 

167-169 

316. Demonstration of the principle of least 

action, . . . . 167-168 

317. Maupertius's a priori deduction of 

the principle of least action, . 168 

318. Deduction of the dynamical equa- 

tions from the principle of least 
action, .... 168-169 

319-322. Principal Function and 

other similar Functions, 169-170 

319. Hamilton's principal function and 

its use (169 9 ), . . . .169 

320-322. Other functions suggested by 



XX11 



ANALYTICAL TABLE OF CONTENTS. 



Hamilton, instead of the charac- 
teristic function (169 2C ), (170 13|24 ), 

169-170 

323-325. Partial Differential Equa- 



tions for the Determination 
of the Characteristic, Prin- 
cipal, and other Functions 
of the same Class (171 18 _ 22 ), 
(171^-1720, .... 171-172 



CHAPTER X. 



integration of the differential equations of motion. 



326. Jacobi's discussion of differential 
equations important to a full de- 
velopment of the problem of me- 
clianics, 172 



I. Determinants and Functional De- 
terminants. 

327. Gauss's determinants (173 12 ), . 172-173 

328. Reversal of the sign of the determi- 

nant, 173 

329. The equality between the elements 

for which the determinant van- 
ishes (173 29 ), .... 173-174 

330. Eeduction of the forms of the deter- 

minant when certain elements van- 
ish (174 14 , ^3), . . . .174 

331. An element can- be taken out as a 

factor when all the elements of a 
certain class vanish (1742,,, 31 ), . 174 

332. Two elements can be taken out as a 

factor, by an extension of the pre- 
ceding principle (175 3|G ), . .175 

333. By the ultimate extension of this 

principle, the determinant is re- 
duced to the continued product of 
its leading terms (175 10jl3 ), . .175 

334. The complete determinant expressed 

by means of the partial determi- 
nants (175 21]27 ), 175 

335. Deduction of all the partial determi- 

nants from one, . . . 175-176 

336. Conditional equations for the par- 

tial determinants (136 10>13 ), . . 176 

337. Expi-essions of the determinant and 

of the partial determinant by par- 



tial determinants of the second or- 
der (136 20 ,„ 7 ), . . . .176 

338. Mutual relations of the partial deter- 

minants of the "second order, and 
corresponding reduction of the ex- 
pression of the complete determi- 
nant (177 12il5 ), . . . 176-177 

339. The solution of linear equations by 

the aid of determinants (177 21j31 ), . 177 

340. Ratios of the unknown quantities 

when the second members vanish 
(178 12 ), 178 

341. 342. Determinants found from the 

partial determinants taken as ele- 
ments (179 31 ), (180 8 ), . . 178-180 

343. The variation of a function of the 

elements, and of the determinant 
(180 13 , 25 ), 180 

344. The variation in a special case of 

symmetrical elements (180^ 31 ), 
(181 ), 180-181 

345. Inverse solution of equation, with the 

corresponding variations (181 n _ M ), 181 

346. Determinant of compound functions 

of two systems of elements (181,3), 
(182^), 181-182 

347. Cases in which the number of com- 

pound elements is not more than 
equal to that of the simple elements 
of each system (182 18 , 20), . .182 

348. Case in which the compounded sys- 

tems are identical (183^ 9 ), . 182-183 

349-373. Functional Determinants, 

183-198 

349. Definition of functional determinant, 



ANALYTICAL TABLE OF CONTENTS. 



XX111 



and its relation to previous propo- 
sitions (183 20 ), . . . .183 

350. Case in which the functional deter- 

minant is the product of two de- 
terminants (183 31 ), . . . 183 

351. Case in which the functional deter- 

minant is a continued product of 
derivatives (184 6 ), . . .184 

352. The determinant of mutually de- 

pendent functions vanishes, . .184 

353. A dependent function is constant in 

finding the functional determinant, 

184-185 

354. Simplification of the functional deter- 

minant by successive substitution, . 185 

355. The functional determinant of inde- 

pendent functions does not vanish, 185 

356. The functional determinant of com- 

pound functions, . . . 185-186 
35 7. The inverse and direct function- 
al determinants are reciprocals 
(187 , 10 ), .... 186-187 

358. Relation of inverse derivative to par- 

tial functional determinant (187 28 ), 187 

359. Variation of functional determinant 

(188,), 187-188 

360. Variation of inverse functional de- 

terminant (188,,), . . . .188 

361. Mutual relation of the partial func- 

tional determinants and variations 

of the functions (188 23 ), . .188 

362. Transformed expression of the func- 

tional determinant (189 8 ), . 188-189 

363. 364. Functional determinant of im- 

plicit functions (189.2s), (190^), 189-190 

365. Determinant of partial functional de- 

terminants (191 5 ), . . . 190-191 

366. Determinant of partial functional de- 

terminants of inferior order (191 25 ), 191 

367. 368. Determinants of mixed partial 

functional determinants (192 10il4 ), 

191-192 
369, 370. Sum of products of mixed par- 
tial determinants (193 22 i), • 192-193 

371. Lagrange's equations for determi- 

nant of partial derivatives (194 5 ), 

193-194 

372. Substitution of the functional deter- 



minant of the function or its higher 
derivatives for the first derivative 
(194 31 ), 194-195 

373. Determination of a system of func- 

tions for which the functional de- 
terminant is given (192 192; ), . . 195 

374-375. Multiple Derivatives and 

Integrals, .... 196-198 

374. Transformation of multiple deriva- 

tives from one set of variables to 
another by means of determinants 
(197 10 ), 196-197 

375. Multiple integrals transposed into a 

sum of linear integrals (198 19 ), 197-198 



II. Simultaneous Differential Equations, 
and Linear Partial Differential 
Equations of the First Order. 

376. An integral of simultaneous differen- 

tial equations (199 19 ), . . . 199 

377. The solution of a linear partial dif- 

ferential equation (200 2 ), . 199-200 

378. Relation of the integral of simulta- 

neous differential equations to the 
solution of a linear partial differen- 
tial equation, .... 200 

379. Transformation of linear partial dif- 

ferential equations so as to reduce 
the number of variables, . . 200 

380. A solution can always be obtained 

by series (201 22 ), . . . 200-201 

381. The number of independent solu- 

tions of a linear partial differential 
equation, .... 201-203 

382. Any function of the solutions is a so- 

lution, 203 

383. General and particular systems of 

integral equations (203 25 , 3 i), . . 203 

384. Each equation of a general system of 

integral equations is an integral, . 204 

385. Relations of a particular system to 

the integrals, .... 204 

386. That portion of the equations of a 

particular system which does not 
involve arbitrary constants is itself 
a particular system, . . . 204 



XXIV 



ANALYTICAL TABLE OF CONTENTS. 



387. The conditional equations to which 

the ai'bitrary constants of a general 
system must be subject, for a par- 
ticular system (203 3X ), . . 204-206 

388. Process of deriving a system of in- 

tegral equations from one of its 
components, . . . . . 206 

389. Test that a proposed equation is not 

an integral, ..... 206 

390. Superfluous constants lead to new 

integral equations (20 7 13 ), . 206-208 

391. The system of integral equations in 

which the initial values are the ar- 
bitrary constants, .... 208 

392. An integral equation in which the 

initial values are the arbitrary 
constants may be changed to an- 
other integral equation in which 
the initial values are the varia- 
bles, 208-209 

393. The integral equation which expresses 

the value of an initial value of the 
variable, transformed to one which 
expresses the value of the variable 
(209 1S ,, 8 ), 209 

394. Differential equations of high orders 

reduced to the first order when 
they are given in the normal form 
(210 I4 _ 17 ), .... 209-210 

395. Reduction of differential equations 

to the normal form, . . 210-211 

396. Case in which the order of differen- 

tial equations admits of reduction 

211-212 

397. One normal system transformed to 

another, 212-213 

398. Normal systems transformed so as to 

contain only two variables, . 213-214 

399-431. The Jacobian Multiplier 
of Differential Equations, 

214-231 

399. Definition of Jacobian multiplier, . 214 

400. The functions of the multiplier are 

the independent solutions, . .214 

401. The linear partial differential equa- 

tions by which the multiplier is de- 
fined (215!,), .... 214-215 



402. The common differential equation 

which defines the multiplier (216 2 ), 

215-216 

403. The multiplier expressed by a deter- 

minant (216 u ), . . . .216 

404. The multiplier expressed by a deter- 

minant of implicit functions (216 17 ), 216 

405. The multiplier expressed by the in- 

verse determinant (21630), . 216-217 

406. The ratio of two multipliers is a so- 

lution (217 15 ), . . . .217 

407. Case in which unity is a multiplier 

(217 27 ), 217-218 

408. Determination of the multiplier when 

its solutions are known, . . 218 

409. Determination of the multiplier when 

the solutions have the forms of the 
initial values of the variables, . 218 

410. Case of greatest simplicity in the de- 

termination of the multiplier, 218-219 

411. Substitution of the arbitrary con- 

stants for their equivalent functions, 219 

412. Reduction of the form when one of 

the variables is a solution, . 219-220 

413. Transformation of the multiplier with 

the change of variables (211 n ), 220-221 

414. Corresponding transfoi'mation of the 

equations for the determination of 
the multiplier (22L„), . . .221 

415. Transformation when the element of 

differentiation remains unchanged 
(261 2M1 ), • • • . 221 

416. Transfoi'mation with a partial change 

of variables (222 8 ), . . . 222 

417. Transformation when the common 

element of differentiation is a va- 
riable, 222 

418. Transformation when part of the 

new variables are solutions, and 
the multiplier is unchanged (222 25 ), 222 

419. Transformation when part of the 

new variables are solutions and the 
element is unchanged, . . 222-223 

420. Transformation when all the new 

variables are solutions (223 8 ), . 223 

421. The multiplier of differential equa- 

tions of a higher order expressed in 
the normal form (222 2S ), . . 223 



ANALYTICAL TABLE OF CONTENTS. 



XXV 



422. Case in which the multiplier of dif- 

ferential equations of a higher or- 
der is unity (223*,), . . . 223 

423. Equation for the multiplier of differ- 

ential equations of a higher order 
when they are not in the normal 
form (224 31 ), 224 

424. Case in which the equation for the 

multiplier admits of reduction 
(225 4 , 10 ), 225 

425. Case in which the given equations 

cannot be reduced to the normal 
form without differentiation, . . 225 

426. Direct determination of the functions 

involved in the equation of the mul- 
tiplier from the given equations, 

225-226 

427. Determination of the factor for the 

passage from the multiplier of the 
given equation to that of one of 
the simplest forms of normal equa- 
tions (228 8 ), .... 226-228 

428-431. Principle of the Last Mul- 
tiplier, .... 228-231 

428. The Jacobian multiplier coincides 

with the Eulerian multiplier when 
there are two variables (229 2 ), 228-229 

429. Jacobi's principle of the last mul- 

plier, 229 

430. By the principle of the last multiplier, 

when the element of variation is not 
directly expressed in the given equa- 
tions, either the two last integrals 
can he obtained by quadratures, or 
the last integral can be obtained with- 
out integration, .... 229 

431. The principle of the last multiplier 

when a portion of the variables are 
not involved in the remaining de- 
rivatives, and the remaining de- 
rivatives satisfy a given equation, 

230-231 

4a2-441. Partial Multipliers, . 231-235 

432. Definition of the partial multipliers 

(231 15 ), 231 

d 



433, 434. Defining equation of the partial 

multiplier (231. 28 ), (232 3 ), . 231-232 

435. Determination of the signs in the for- 

mation of the multipliers (23 2 1:l ), . 232 

436. Case in which the partial multiplier 

is the Jacobian multiplier, . . 232 

437. Case in which the partial multiplier 

is the Eulerian multiplier amplified 
by Lagrange, . . . .232 

438. Every partial multiplier corresponds 

to an integral of the equation, . . 233 

439. The deduction of an integral from 

the Eulerian multiplier (233 27 ), 233-234 

440. Transformation of the partial multi- 

plier when there is a change of va- 
riables (239 27 ) . . . .239 

441. Transformation and reduction of 

the partial multiplier when the 
solutions are adopted as new vari- 
ables, .... 234, 235 

III. Integrals of the Differential Equa- 
tions of Motion. 

442. General integrals of the equations of 

motion, 235 

443-451. The Application of Ja- 
cobi's Principle of the Last 
Multiplier to Lagrange's 
Canonical Forms, . 236-241 

443. Lagrange's canonical forms consti- 

tute a system of normal forms 236 

444. A Jacobian multiplier is always 

known in equations of motion when 
the forces do not involve the veloc- 
ities (237 18 ), . . . 236-237 

445. The principle of the last multiplier ex- 

pressed as a dynamical principle, . 237 

446. The Jacobian multiplier when the 

equations of motion are expressed 
in rectangular coordinants (238 13 ), 

237-238 

447. The conditional equations expressed 

in the multiplier of the equations 

of motion (23920), . . . 238-239 

448. The transformation of the multiplier 

by the introduction of the original 



XXVI 



ANALYTICAL TABLE OF CONTENTS. 



elements instead of the rectangular 
coordinates (440 8 ), . . . 239-240 
449. The Jacobian multiplier of the equa- 
tions of motion when there are no 
equations of condition ; it is unity 
when the coordinates are rectangu- 
lar (240 16 ), . . . . .240 



450. The equations of condition considered 

as forces in the expression of the 
multiplier (24 1 7 ), . . . 240-241 

451. The multiplier is unity when the dif- 

ferential equations of motion are 
expressed in Hamilton's form, . 241 



CHAPTER XI. 

MOTION OF TRANSLATION. 



452. 



453. 



The motion of the centre of gravity 
is independent of the mutual con- 
nections, .... 241-242 

The motion of the centre of gravity 
depends upon the external forces, 242 



454-752. Motion of a Point, . 242-433 

454. The differential equations of the mo- 

tion of a point (243 4 ), . . 242-243 

455-459. A Point moving upon a 

Fixed Line, . - . . 243-244 

455. By the principle of the multiplier, the 

motion is expressed by integrals by 
quadratures (243 19|24 ), • • • 243 

456. The velocity dependent solely upon 

position, and not upon the interme- 
diate path, .... 243-244 

457. Case in which the motion is limited, 

in which case the oscillations are 
invariable in duration, . . . 244 

458. If the path returns into itself, the 

period of circuit is constant, . . 244 

459. Expression of the multiplier when 

the forces and equations of motion 
involve the time (244 28 ), . . 244 

460-477. The Motion of a Body upon 
a Line when there is no Ex- 
ternal Force. Centrifugal 
Force, .... 245-254 

460. Upon a fixed line with no external 

force the velocity is constant, . 245 

461. Measure of the centrifugal force 

(245 17 ), 245 



462. 
463. 
464. 
465. 

466. 

467. 
468. 
469. 

470. 
471. 

472. 

473. 



474. 



Total pressure upon a line where 
there are external forces, . . 245 

The centrifugal force cannot be used 
as a motive power, . . . 245 

The acceleration of a body upon a 
moving line (246 26 ), . . 245-247 

Upon a uniformly moving line the 
relative velocity of a body acted 
upon by no force is constant, . 247 

The acceleration of a line moving 
with translation is a negative force 
acting upon the body, . . .247 

The same proposition applies to any 
line, 247-248 

Case in which the line rotates uni- 
formly about a fixed axis (248 24 ), . 248 

The time of oscillation of a body up- 
on a uniformly rotating line is 
constant, .... 248-249 

The period of circuit of a body upon 
a uniformly rotating line is constant, 249 

Case in which the motion of the body 
vanishes at the axis of rotation 
(249 20 ), 249 

Motion of a body on a uniformly 
rotating straight line (250 6|2 o, 3j)j 
(251 2 ), 249-251 

Motion of a body on a uniformly ro- 
tating circumference of which the 
plane is perpendicular to the axis 
of rotation (251 29 ), (252 3 22 26 ), 
(253 3 , n , 21 ), .... 251-253 

Motion of a body upon a rotating 
line which is wholly contained up- 



ANALYTICAL TABLE OF CONTENTS. 



XXV11 



on the surface of a cylinder of revo- 
lution of which the axis is the axis 
of rotation (2532s), . . .253 

475. Case in which the rotation of the cyl- 

inder is uniform, .... 254 

476. Case in which the curve is a helix 

(254 8 ), 254 

477. Case in which the acceleration is 

uniform (254i 5 ), .... 254 

478-482. Motion of a Heavy Body 
upon A fixed Line. The Sim- 
ple Pendulum, . . . 254-256 

478. The motion of a heavy body upon a 

fixed line (254 28 ), . . . .254 

479. When the line is contained upon the 

surface of a vertical cylinder, 254-255 

480. When the line is straight (255 G ), . 255 

481. When the line is straight and no ini- 

tial velocity (255 13 ), . . . 255 

482. When the line is the circumference 

of a circle ; oscillations of the sim- 
ple pendulum (255 25 ), (256 3;16)20 ,«5), 

255-256 

483-502. Motion of a Heavy Body 

upon a moving Line, .257-270 

483. When the line has a motion of trans- 

lation (25 7 8 ), . . . .257 

484. When the translation is uniformly 

accelerated, it is equal to a constant 
force, 257 

485. When the line is straight, and the 

law of translation is given ; in 
what case this path is a parabola 
(257^), (258,, „), . . 257-258 

486. When the translation of the line is 

uniform and direct ; gain of power 
(25830), (259 16 _ 25 ), . . . 258-259 

487. When the line is the circumference 

of a vertical circle (259 31 ), (260 19>24 ), 
(261 6 , 8 , H , 18 ), .... 259-261 

488. When the line rotates about a verti- 

cal axis (26125), . . . .261 

489. When the line rotates uniformly 

about a vertical axis (262 2 ), . 261-262 

490. When a straight line rotates uni- 



formly about the vertical axis 
(262 13 ), 262 

491. Direct integration of the linear dif- 

ferential equation in this case into 
the form given by Vieille (262 19 ), 262 

492. Case of § 490, in which there is an 

impassable limit (26225,3!), (263„), 

262-263 

493. Case of § 490, in which there is no 

limit (263 21J1 ), . . . .263 

494. Case of § 490, in which there is a 

possible position of immobility 
(264,, w ), 264 

495. When the circumference of a circle 

rotates uniformly about a vertical 
axis ; the point of maximum and 
minimum velocity defined by an 
hyperbola (265J2), . . . 264-265 

496. Case of § 495, in which there is no 

motion upon the line, . . . 266 

497. Case of § 495, when the minimum 

velocity vanishes (26 7 21 ), . 266-267 

498. When a parabola of a vertical trans- • 

verse axis rotates uniformly about 

its axis (268 10>31 ), . . . 267-269 

499. Case of §498, when the minimum 

velocity vanishes (269 u ), . . 269 

500. When the axis of rotation is not ver- 

tical, and when the rotation is uni- 
form (269 18 ), 269 

501. When a straight line rotates uniform- 

ly about an inclined axis (270 6 ), 

269-270 

502. Rotation of a plane curve about an 

inclined axis (270 13 ), . . . 270 

503-534. Motion of a Body upon a 
Line in opposition to Fric- 
tion, OR THROUGH A RESISTING 

Medium, .... 270-315 

503. The resistance of a medium, . .270 

504. Expression of the resistance, . 270 

505. Resistance of a medium to the mo- 

tion of a body upon a fixed line 
(271 3 , 9 ), 271 

506. Motion of body upon a fixed line 

through a resisting medium with- 
out external force (271 12) i 7 ), . . 271 



xxvm 



ANALYTICAL TABLE OF CONTENTS. 



511. 



512. 



507. Case of § 506, when the law of resist- 

ance is expressed as a quadratic 
function of the velocity. Change 
of sign of the resistance (271 27 ), 
(272 4 , 13 , 22 ), .... 271-273 

508. Case of § 506, when the resistance 

is friction upon the line (273 10 ), . 273 

509. Case of § 508, when there is no ex- 

ternal force (27328), . . .273 

510. Case of § 509, when the fixed line is 

the involute of the circle (274 6 ), . 274 
Case of § 509, when the line is the 

logarithmic spiral (274 14 ), . .274 
Case of § 509, when the line is the 

cycloid (2742o), . . . .274 

513. Case of § 506, when the resistance of 

the line is constant, and the resist- 
ing medium moves with a uniform 
velocity, and the resistance is pro- 
portional to the velocity (275 3 ), 274-275 

514. Case of §513, when the line is 

straight, and there is no external 
force (275 n , 2D ), . . . 275-276 

515. Motion of a heavy body upon a fixed 

straight line, when the resistances 
are friction, and that of a moving 
medium which resists as the square 
of the velocity (276 31 ), (277 8>20 ), 
(278 16 ), (279 3 , 15 , 28 ), (280 Wia) ), 276-280 

516-534. The Simple Pendulum in 

a resisting medium, . 281-315 

The small oscillations of a pendulum 
against friction and the resistance 
of a medium which is proportional 
to the velocity (281 20 _ 23 ), . . 281 

The oscillation after many vibra- 
tions (282 4 ), • • • " • 282 

The time of oscillation compared 
with that in a vacuum (282 18|23 ), . 282 

The arc of oscillation (283 2 ), . 282-283 

The law of diminution of the arc of 
oscillation and of the maximum of 
velocity (283 28 ), . . . .283 

The oscillations of the pendulum 
if the resistance is as the square of 
the velocity (284 23 , 31 ), . . 284-285 



516 



517. 

518. 

519. 
520. 

521. 



522. The arc of oscillation in the case of 

§ 521 (285 20 ), . . . .285 

523. The arc of oscillation in the case of 

§ 521 is the same as in a vacuum 
(286 12>17 , 22 ), .... 285-286 

524. The oscillations of the pendulum 

when the law of resistance is ex- 
pressed as a function of the time 
(287^), .... 286-287 

525. The oscillations of the pendulum, 

as affected by those produced in 
the medium (28822-24), (289 24 _26), 
(2902o), (291 4 ), . . . 287-291 

526. The oscillations of the pendulum as 

affected by the portion of the me- 
dium which becomes part of the 
pendulum (291 28 _ 3 i), . . 291-292 

527. Constants of the formula? of the oscil- 

lations of the pendulum in a resist- 
ing medium arranged for applica- 
tion to experiment (292 17 . 19 ), . . 292 

528. Approximate form for the best exper- 

iments in which the friction is in- 
sensible (292 24 _2 8 ), • • • .292 

529. The French system of weights and 

measures adopted in the examina- 
tion of experiments, . . 292-293 

530. Discussion of Newton's experi- 

ments upon the pendulum in air 

f ^JtJj^oo 2G-2&)? ..." £<JO ZJQ 

531. Discussion of Dubuat's experi- 

ments upon the pendulum in air 
and water (295 6 _ 7> 13 _ 1G ), . . 294-295 

532. Discussion of Borda's experi- 

ments upon the pendulum in air 
(296 M ,„_ 13 , 17 _ 18 ), .. . • 296-297 

533. Discussion of Bessel's experi- 

ments upon the pendulum in air, 
(298 7 _ 9> 13 _ 15j 2o_22, 25-27)) (299 n _ 13i 15.17), 
(299 20 „22, 20-27), • • • • 298-311 

534. Discussion of Baily's experiments 

upon the pendulum in air, . 311-315 

535-559. The Tautochrone, . 316-327 

535. Definition of the tautochrone, . .315 

536. The case of the tangential force of 

the tautochrone when it can be ex- 



ANALYTICAL TABLE OF CONTENTS. 



XXIX 



pressed as a function of the arc 
(317 D ), 316-317 

537. The equation of the tautochrone un- 

der the action of a fixed force 
(317 1C ), 317 

538. The tautochrone which rotates uni- 

formly about a fixed axis when 
there is no external force (31 7 25 ), . 317 

539. The case of § 538, when it is a plane 

curve (318 4 , 7 ), . . . 317-318 

540. The cycloid is the tautochrone of a 

free heavy body in a vacuum 
(318 28 ), 318-319 

541. The tautochrone of a heavy body in 

a vacuum upon a given surface 
(319 13 ), 319 

542. The tautochrone of § 541, when the 

surface is a cylinder of which the 
axis is horizontal, and the equa- 
tion of the base is (319 25 ), (320 5 ), 

319-320 

543. The tautochrone of § 542 upon the 

developed cylinder (320 14 ), . . 320 

544. The tautochrone of § 542, when it 

passes through the lowest side of 
the cylinder (3 20 I4 , 19 , 24 ) , . 320-321 

545. The differential equation of the tau- 

tochrone of § 542 referred to rec- 
tangular coordinates (321 10>18 ), . 321 

546. The tautochrone of § 542, when the 

base of the cylinder is a cycloid 
(321„,), (322 5 ), . . . 321-322 

547. The tautochrone of a heavy body 

upon a surface of revolution of 
which the axis is vertical, and the 
meridian curve is that of (319 25 ), 
(322 19 ), 322 

548. The tautochrone of a heavy body 

upon a vertical cone of revolution 
(322 28 ), (323 :i ), . . . 322-323 

549. The tautochrone of §548, which 

passes through the vertex (323 9 ), . 323 

550. The tautochrone of § 547, when 

the meridian curve is a cycloid 
(323 13 ), 323 

551. The tautochrone upon a plane when 

the force is directed towards a 
point in the plane, and propor- 



tional to some power of the dis- 
tance from the point (323, 5 ), . 323 

552. The tautochrone of § 551, when the 

force is any function of the dis- 
tance (324 16 ), .... 323-324 

553. The polar differential equation of the 

tautochrone in the case of § 552 
(324^), 324 

554. The differential equation of the tau- 

tochrone of § 552 in terms of the 
radius of curvature and the angle 
of direction (324 27 ), . . .324 

555. The tautochrone of § 552, when it is 

the involute of the circle (325 n ), . 325 

556. The tautochrone of § 552, when it 

is a logarithmic spiral (325 2G ), 

325-326 

557. The tautochrone of §552, when the 

force is proportional to the dis- 
tance from the origin ; when it is 
not infinite, it is an epicycloid 
(326 21 ), (327 10 ), . . . 326-327 

558. Cases included in § 557, near the 

point of greatest velocity, . . 327 

559. The tautochrone in a resisting me- 

dium postponed to case of holo- 
chrone, 327 

560-604. The Brachistochrone, 328-354 

560. Definition of the brachistochrone, . 328 

561. The investigation of the free brachis- 

tochrone (328 10 ), . . . .328 

562. The brachistochrone when. the act- 

ing forces are fixed (328 24|28 ), . 328 

563. The pressure upon the brachisto- 

chrone is double the centrifugal 
force (329 4 ), .... 328-329 

564. The point of contrary flexure in a 

brachistochrone, .... 329 

565. The conditions of the brachisto- 

chrone introduced by the general 
method of variations, . . . 329 

566. When the force is directed towards 

a point, the free brachistochrone 
is a plane curve, and its plane in- 
cludes the point of attraction, . 329 

567. When the forces are parallel, the 

free brachistochrone is a plane 



XXX 



ANALYTICAL TABLE OF CONTENTS. 



568. 



curve, and its plane is parallel to 
the direction of the forces, . . 329 
When there are no forces the brachis- 
tochrone is the shortest line, . 329-330 

569. The equation of the brachistochrone 

when its force is central (330 I5 ), . 330 

570. The brachistochrone of § 569, when 

the force is proportional to the dis- 
tance ; it is a spiral or an epicy- 
cloid (331 13 , a,), . . . 330-331 

571. The equation of the brachistochrone 

when the forces are parallel (331 31 ), 

331-332 

572. The brachistochrone of a heavy body 

is a cycloid (332„), . . .332 

573. The brachistochrone of § 571, when 

the force is proportional to the dis- 
tance from a given line (332 19 ), 
(333 12 ), .... 332-333 

574. The centrifugal force in the brachis- 

tochrone upon a given surface, . 333 

575. Simple case of a brachistochrone 

upon a given surface, including 
that of a meridian line upon a sur- 
face of revolution, . . 333-334 

576. The brachistochrone upon a surface 

of revolution when the force is di- 
rected to a point of the axis, and 
expression of the projection of the 
area upon the plane perpendicular 
to the axis (334 2S _ 29 ), . . 334- 

577. The derivatives of the arc and of the 

difference of longitude in the case 
of § 576, taken with reference to 
the arc of the meridian (335i_ 7 ), . 

578. The derivatives of the same quanti- 

ties taken with reference to the 
latitude (331„_ 19 ), .... 

579. The surface upon which the brachis- 

tochrone may make a constant 
angle with the meridian ; it may 
be used to define the limits of the 
brachistochrone in any case of 
§ 576 (33522), (336 9 ), . . 335-336 
The limiting surface of § 579 is a 
paraboloid of revolution in the case 
of a heavy body, of which the axis 
is directed downwards. Investiga- 



335 



335 



335 



580. 



581 



582. 



583. 



584, 



585, 



586. 



587. 



588. 



589. 



590. 



591. 



tion of the other brachistochrones 
upon this surface (336 29 ), (337 8 , 17? 27 ), 
(338 2 7 13> 19 _ 20 ), (339 5i 13j 20 ), (340 7i 14 ), 

336-340 

The brachistochrone for a heavy 
body upon a paraboloid of revolu- 
tion of which the axis is the upward 
vertical (340 21 ), (341 8 ), . . 340-341 

The brachistochrone of a heavy body 
upon a vertical right cone (341 20 ), 
(342 2 _ 5 , 10 , 23 ), (343 2 , 13 , 19 , 22, 25), (344 2 ), 

341-344 

The brachistochrone of a heavy body 
upon an ellipsoid of revolution of 
which the axis is vertical (344 14 ), 
(345 19 , 25 ), (346 M , 14 _ 19>25 ),. . 344-346 

The tangential radius of curvature of 
the brachistochrone of a heavy body 
upon any surface (34 7 7 ), . 346-347 

When the force is parallel to the axis, 
and proportional to the distance 
from a plane which is perpendicular 
to the axis, the limiting surface of 
§ 579 is an ellipsoid or an hyper- 
boloid, ..... 

When the force is proportional to the 
distance in § 576, the limiting sur- 
face of § 579 is an ellipsoid or an 
hyperboloid, 

Investigation of the limiting surface 
of § 579 when the force is propor- 
tional to the square of the distance 
in § 576, .... 

The normal pi-essure upon the brach- 
istochrone when the length of the 
arc is given (34 7 29 ), . .347-348 

The equation of the brachistochrone 
in the case of § 569, when the length 
of the arc is given (348 cln ), . 

The equation of the brachistochrone 
in the case of parallel forces when 
the length of the arc is given 
(348 14 , 18 ), .... 

The equation of the brachistochrone 
in the case of § 576 when the 
length of the arc is given. The 
investigation of the limiting sur- 
face (348 25 , 29 ), (349,), . . 348-349 



347 



347 



347 



348 



348 



ANALYTICAL TABLE OF CONTENTS. 



XXXI 



349 



349 



592. The normal pressure in a brachisto- 

chrone when the total expenditure 
of action is given (349 10 ), 

593. The equation of the brachistochrone 

in the case of § 576, when the to- 
tal expenditure of action is given 

594. The equation of the brachistochrone 

in the case of parallel forces when 
the total expenditure of action is 
given (349 31 ), (350 2 ), . . 349-350 

595. The equation of the brachistochrone 

in the case of § 576, when the to- 
tal expenditure of action is given, 
and the investigation of the limit- 
ing surface (350 S1 ., 1S ), . . . 350 

596. The brachistochrone in a medium of 

constant resistance (351 3 _<, il3 ), 350-351 

597. The expression of the multiplier of 

the equation for the element of 
length of the arc when the force is 
central in the case of § 596 (351 20 ), 351 

598. The expression of this multiplier 

when the forces are parallel (351 24 ), 351 

599. The equation of the brachistochrone 

of a heavy body in a medium of 
constant resistance (351 29 ), . . 351 

600. The brachistochrone in a medium, 

of which the resistance is a given 
function of the velocity (352i 6 _ 18 , 25 ), 352 

601. The equations of § 600 when the 

forces are parallel (352 29 _ 31 ), . . 352 

602. The brachistochrone of a heavy body 

in any resisting medium, and in the 
case of the resistance inversely pro- 
portional to the velocity, and di- 
rectly proportional to the square of 
the velocity, (353^,20,24.26), • 

603. Euler's error in regard to the nor- 

mal pressure of a brachistochrone 
in a resisting medium, . 

604. Singular difficulty in the special de- 

termination of the brachistochrone 
when its form is given. Special 
example of such an investigation 
(354 15 ), 



353 



353 



605. 
606. 



607. 



608. 



609. 



610. 



611. 



612. 



613. 



614. 



615, 



616 



605-624. The Holochrone, 



354 



354-364 



617 



618 



Definition of the holochrone, . . 354 

The force along the curve of the 
holochrone when the forces are 
fixed (355 4 , 9 ), . . . 354-355 

The holochrone for a heavy body 
(355 13 ), 355 

The force along the holochrone when 
the time of descent admits of de- 
velopment according to integral 
ascending powers of the arc 
(355 18 ^ 1 ), ..... 355 

Given function of the initial value of 
the potential (35 7 9 , 15 ), . . 356-357 

Case of § 609 when the forces are 
parallel (3572,), . . . .357 

Case of § 609 when the forces are 
central (3572,), . . . .357 

Case of § 609 when the time is devel- 
oped according to powers of the 
initial value of the potential (358 6 ), 358 

Case of § 609 when the curve of ap- 
proach to the point of maximum 
potential is given, and the whole 
time is a given function of the 
maximum potential (358 17 ), . . 358 

Case of § 609 when the time of os- 
cillation is constant, which is a cu- 
rious species of tautochrone inves- 
tigated by Euler for heavy bodies 
(358 27 ), 358 

The holochrone is indeterminate 
when the forces may depend upon 
the velocity, but there is a con- 
dition which must be satisfied 
(359 ln ), .... 358-359 

The case of § 615 when the force 
along the curve may be separated 
into two parts, of which one is fixed 
and depends upon the arc, while 
the other depends upon the veloc- 
ity, has been largely discussed with 
little success and much animosity . 359 
The case of§ 615 transformed to La- 
grange's most general form, of the 
tautochrone (359 2? ), . . . 359 

The case of § 615 transformed to La- 
place's general form of the tauto- 
chrone (360 6 ), . . . .360 



XXX11 



ANALYTICAL TABLE OF CONTENTS. 



619. The case of § 615 transformed to a 

third form equally general with 
those of Lagrange and Laplace 
(360 14 ), 360 

620. The case of § 615 when the assumed 

equation consists of three parts, 
which are functions respectively 
of the arc, the time, and the veloc- 
ity (360 27 ), 360 

621. Transformation of the preceding form 

to a familiar formula of La- 
grange (36 1 2 ), . . . 360-361 

622. Case in which the form of § 621 co- 

incides with that of §616, which 
sustains the correctness of Fon- 
taine's strictures ; and this holo- 
chrone is essentially tautochronous 
(362 3 ), 361-362 

623. Case of § 615, which includes La- 

grange's formula (362 31 ), (363 6 ), 

362-363 

624. Case of §621, in which the force 

along the curve has the form given 
in §616, with the addition of a 
term which is the product of the 
square of the velocity by a function 
of the arc (363 31 ), . . .363-364 

625-639. The Tachytrope, . . 364-368 

625. Definition of the tachytrope, . . 364 

626. Application of §615 to the tachytrope, 364 

627. Case in which the time is not in- 

volve! in the assumed equation of 
§ 615, and the force has the form 
of §616 (364 16 ), .... 364 

628. Case of a heavy body in which the 

tachytrope is a cycloid (364 23 , a), 
(365 4 ), 364-365 

629. Klingstierna's case of the tachy- 

trope in a medium which resists as 
the square of the velocity, which 
was solved by Clairaut (365 21 ), 365 

630. Case of § 627, when the velocity is 

uniform (365.,-,), .... 365 

631. Case of the tachytrope when the 

forces are parallel, and the as- 
sumed equation of §615 does not 
involve the time (366 4 ), . . 366 



632. Case of §631, when the velocity 

has a constant ratio to that in a 
vacuum (366 1219 ), . . . . 366 

633. Case of §627, when the forces are 

central, and the assumed equation 
is expressed in terms of the veloc- 
ity and the radius vector (366 24 ), . 366 

634. Case of §633, when the velocity has 

a constant ratio to that in a vacuum 
(366 31 ), (367 3 ) r . . . 366-367 

635. Case in which the velocity in a given 

direction is a given function of the 
arc and the distance in that direc- 
tion (367 12 ), 367 

636. Case in which the velocity in a given 

direction is uniform (36 7 19 ), . . 367 

637. Case of § 636 for a heavy body 

(367 25j31 ) 36 7 

638. Case of § 637 when there is no re- 

sisting medium ; for a horizontal 
direction the tachytrope is a para- 
bola, and for a vertical direction 
it is the evolute of a parabola 
(368 3 , 8 , 15 , 22 ), 368 

639. The tachytrope of a heavy body when 

the resistance is proportional to the 
velocity (368 31 ), . . . .368 

640-646. The Tachistotrope, . 369-370 

640. Definition of the tachistotrope, . 369 

641. The tachistotrope in a medium of 

which the resistance is a given 
function of the velocity (369 14 _ 21 ), . 369 

642. The normal pressure on the tachis- 

totrope when the resistance is pro- 
portional to a power of the veloc- 
ity (369 27 ), 369 

643. The tachistotrope is a straight line 

when the resistance is constant, . 370 

644. The tachistotrojie for parallel forces 

(370 5 ), 370 

645. The tachistotrope of a heavy body 

(370 10 _ 12 ), 370 

646. The tachistotrope of a heavy body 

when the resistance is that of 
§642(370 1C ), . . . .370 

647-655. The Barytrope and the 

Tautobrayd, . . , 3 70-373 



ANALYTICAL TABLE OF CONTENTS. 



xxxm 



647. Definition of the barytrope andtauto- 

baryd, 370 

648. The barytrope when the force has 

the form of § GIG (370 31 ), . .370 

649. The barytrope and tautobaryd of a 

heavy body (371 30 ), . . .371 

650. The barytrope and tautobaryd when - 

the resistance is constant; this in- 
vestigation is applied to a heavy 
body (371 13 , 17i20 , 31 ), (3 72.,), . 371-372 

651. The barytrope against which there 

is no pressure is the curve of free 
motion (372 10 ), . . . .372 

652. When the curve of the bar) r trope is ' 

given, the relations of the fixed 
force and resistance, . . .372 

653. These relations in the case of parallel 

forces (372 19i24 ), . . . .372 

654. The relations of §653 applied to the 

circle (372 29 ), (373 3 ), . . 372-373 

655. The relations of § 653 applied to a 

cycloid (373io 16 ), . . . .373 

656-662. The Synchrone, . . 373-374 

656. Definition of the synchrone and its 

dynamic pole, . . . .373 

657. The synchrone for a constant time, 

373-374 

658. The synchrone in a resisting medium 

without force, on a path of given 
form is the surface of a sphere, . 3 74 

659. The synchrone for a uniformly ro- 

tating straight line without exter- 
nal force is a surface of revolution, 3 74 

660. The synchrone for certain fixed 

forces upon straight lines is a sur- 
face of revolution, . . . .374 

661. The synchrone of a heavy body with- 

out resistance (3 74oo), . . . 374 

662. The synchrone of a heavy body in a 

medium which resists as the square 

of the velocity (374 31 ), . . .3 74 

663-670. The Syntachyd, . . 375-376 

663. Definition of the syntachyd, . .375 

664. Investigation of the syntachyd, . 375 

665. In the case of § 658, the syntachyd 

coincides with the synchrone, . 375 

e 



666. In the cases of §§659 and GG0, the 

syntachyd is a surface of revolu- 
tion, .375 

667. When the action is that of fixed 

forces, the syntachyd is a level sur- 
face, . . . . . .375 

668. The syntachyd for a heavy body 

moving upon a straight line against 
a constant friction and through a 
medium of which the resistance is 
proportional to the square of the 
velocity (375 24 ), . . . .375 

669. The syntachyd in a case like that of 

§ 668, but in which the resistance of 
the medium is proportional to the 
velocity (37G 2 ), . . . 375-376 

670. The syntachyd for any body with 

the resistances of § 668 (376 9 ), . 376 

671-735. A Point moving upon a 

Fixed Surface, . . . 376-423 

671. The motion of a point upon a fixed 

surface (377 6 ), . . . 376-377 

672. The centrifugal force of a body 

against a surface (377 l; ), . . 377 

673. When the force is normal to the sur- 

face, the path is the shortest line, . 377 
6 74. When the velocity is constant, the 
body moves upon the intersection 
of the given surface with a level 
surface, . . . . . .377 

675. When the velocity is a given func- 

tion of the parameter of the level 
surface, the equation of a second 
surface upon which the body 
moves, 377 

676. When the force is directed toward 

the origin, the area described by 
the radius vector is proportional 
to the time. The polar equation 
of the path (378 22 ), . . .378 

677. When the force of § 676 is attractive 

and inversely proportional to a 
power of the distance, and the ve- 
locity is that obtained by falling 
from an infinite distance, the polar 
equation of the path admits of sim- 
ple integration ; in gravitation the 



XXXIV 



ANALYTICAL TABLE OF CONTENTS. 



678. 



679. 



path is a parabola ; for a force in- 
versely proportional to the cube 
of the distance, it is a logarithmic 
spiral ; for a force inversely pro- 
portional to the fourth power of 
the distance, it is an epicycloid ; 
for a force inversely proportional 
to the fifth power of the distance, it 
is the circumference of a circle ; for 
a force inversely proportional to 
the sixth power of the distance, it is 
the trifolia; for a force inversely 
proportional to the seventh power 
of the distance, it is the lemniscate ; 
for a repulsive force proportional 
to the distance, it is an equilateral 
hyperbola (379 6 ), . . . 379-380 

Case in which the integration of 
§ 676 is simple (380 20 ), . . .380 

Case of § 6 78 for a force of four terms 
one of which is constant, and the 
others are respectively proportional 
to the distance and to its inverse 
square and cube (381 7jllj2 o, z>, 3i)i 
(382 3;6 ), 380-382 

680. Case of § 678 for a force of four 

terms which are inversely propor- 
tional to the second, third, fourth, 
and fifth powers of the distance 

(382j5 jM? 24, 27, 3V) 

681. Case of § 678 for a binomial form of 

the radical of (378 22 ), (383 5>8 ), 

682. The general forms of force of § 676 

which admit of simple integration 
consist of two terms, of which one 
is inversely proportional to the 
cube of the distance, and the other 
is proportional to the distance or 
inversely proportional to the square 
of the distance (383 13 ), . . .383 

683. The term, which is inversely propor- 

tional to the cube of the distance, does 
not increase the difficulty of integra- 
tion, and the effect of this term may 
he disguised in the constants (383 31 ), 

383-384 

684. Case of no force and of a central 

force inversely proportional to the 



382 



383 



cube of the distance (384 10>2 j i27 ), 
(385 3 , 6 , 10 ), .... 384-385 

685. Case of a central force proportional 

to the distance, . . . 385-386 

686. Case of § 685 combined with a force 

inversely proportional to the cube 

of the distance (38 6 15 _ 18 ), . .386 

687. Case of a central force inversely pro- 

portional to the square of the dis- 
tance, 386-387 

688. Case of § 687 combined with that of 

§ 684 (387 18 ^ 28 ), (388 10 _ 19 ), . 387-388 

689. Case of § 683 with the force of 

§ 684 (388 28 ), . . . .388 

690. The general laws of force for which 

integration may be effected by el- 
liptic integrals, each consisting of 
four terms, with a total variety of 
six cases (389 4 , 8 ), . . . 388-389 

691. Second case of the first form of § 690 

when the force consists of terms 
of the form of § 679 (390 2(M1 ), 
(391 12 _ 14 ), (392 20 ),(393 6 , 1 ^. 23 ),(394 M1 ), 
(394 19 , 24 ), (396 14 ), (397^), (397^), 
(398 8> ,0.2s), (399L0,!,), . . 389-399 

692. First case of the first form of § 690, 

when the four terms are respec- 
tively proportional to the distance, 
to its third and fifth powers, and 
to its inverse cube (399 16|19i 20-31), 
(400 W8 ), .... 399-400 

693. Case of § 692, in which the force is 

proportional to the fifth power of 
the distance, 400 

694. Case of §692, in which the force is 

proportional to the cube of the dis- 
tance, 401 

695. Third case of the first form of § 690, 

in which the four terms are in- 
versely proportional to the cube 
root of the distance, to the fifth and 
seventh powers of the cube root, 
and to the cube of the distance 
(401 15 , 18 , 25 ), 401 

696. Case of §695, in which the force is 

inversely proportional to the cube 
root of the distance, . .401-402 

697. Case of §695, in which the force is 



ANALYTICAL TABLE OF CONTENTS. 



XXXV 



inversely proportional to the fifth 
and the seventh powers of the cube 
root of the distance, . . . 402 

698. Fourth ease of the first form of § 690, 

in which the four terms are inverse- 
ly proportional to the square and 
cube of the distance, and the third 
and fifth powers of the square root 
(40287,2,,), (403.,), . . . 402-403 

699. Cases of § 698, in which the force is 

inversely proportional to the third 
and fifth powers of the square root 
of the distance, .... 403 

700. First case of the second form of § 690, 

in which the four terms are inverse- 
ly proportional to the second, third, 
fourth, and fifth powers of the dis- 
tance (403., 7 ), (404 3iC ), . .403-404 

701. Case of § 700, in which the force is 

inversely proportional to the fourth 
power of the distance, . . . 404 

702. Case of § 700, in which the force is 

inversely proportional to the fifth 
power of the distance, . . 404-405 

703. Second case of the second form of 

§ 690, in which the four terms are 
proportional to the distance, and 
inversely proportional to the third, 
fifth, and seventh powers of the 
distance (405 14 , k,,*.,), • • • 405 

704. Case of § 703, in which the force is 

inversely proportional to the sev- 
enth power of the distance, . 405-406 

705. Third form of central three, in which 

the integration can be performed 
by elliptic integrals (406 1S , 27 , 31 ), . 406 

706. The potential curve for defining the 

limits of the path described under 
the action of a central force (407 10 ), 407 

707. The term of the potential, which cor- 

responds to the force of § 686, may 
be omitted in the potential curve 
of §706, 407 

708. Potential curve in which the path can 

only consist of a single portion, 407-408 

709. The portion of the potential curve 

which corresponds to attraction and 
repulsion, 408 



710. Form of the path for a central force 

in the vicinity of the centre of ac- 
tion, 408-409 

711. Character of the path for a central 

force at an infinite distance from 
the centre of action, . . . 409 

712. Graphic determination of the incli- 

nation of the path to the radius 
vector, 409 

713. The equation of the path for parallel 

forces (410 ), . . . .410 

714. The path of a projectile is a para- 

bola (410 18 ), 410 

715. The equation of the curve for par- 

allel forces referred to rectangular 
coordinates (410 al ), . . .410 

716. The potential curve for parallel 

forces, ...... 411 

717. Case in which the force of §713 is 

proportional to the distance from 

a fixed line (41 1^.^), . . .411 

718. Case in which the force of § 713 is 

proportional to the distance from 
any fixed line divided by the 
square of the distance from an- 
other line (411 26 ), . . . .411 

719. The motion of a body upon a surface 

of revolution when the force is cen- 
tral, and the centre of action is 
upon the axis of revolution (412^), 

411-412 

720. Derivatives of the arc and of the 

longitude in the case of §719 
(412 W2 , 1M7 ), 412 

721. Case of § 719, in which the path of 

the body makes a constant angle 
with the meridian. The surface of 
revolution which defines the limits 
of the path (412 21 ), (413 5 ), . 412-413 

722. Limiting surface of revolution for a 

heavy body (413 n ), . . .413 

723. Motion of a heavy body upon a verti- 

cal right cone (414 3 . ni22 j 1 ), (4155_ 14 ), 
(415 18 . 25i81 ), (416 3 ),. . .413-416 

724. Motion of a heavy body upon a verti- 

cal paraboloid of revolution of which 
the axis is directed downwards 
(416^,28.3!), (417 2>u , 16 ), • 416-417 



XXXVI 



ANALYTICAL TABLE OF CONTENTS. 



725. Motion of a heavy body upon a ver- 

tical paraboloid of revolution of 
"which the axis is directed upwards 
(417^). 417 

726-735. The Spherical Pendulum, 

418-423 

726. The path of a spherical pendulum 

(418* n ), 418 

727. Relation of the limits of the path of 

the pendulum (418 25 _ 27 ), . .418 

728. The time of oscillation for different 

lengths of pendulums, . . 418-419 

729. Case in which the path of the pendu- 

lum is a horizontal circle (41 9 U> 1T| 23) , 
(420 8 , 14 ), .... 419-420 

730. The time of a complete revolution in 

the case of § 729 (420 24 ), (421 4 ), 

420-421 

731. The path of the pendulum when 

it is nearly a horizontal circle 
(421 16|18 ), 421 

732. The path of the pendulum when it is 

nearly a great circle (421 29|31 ), 421-422 

733. The path of the pendulum when it 

passes nearly through the lowest 
point of the sphere (422 10 , 12 ), . 422 

734. Limits of the arc of vibration of the 

pendulum, ..... 422 

735. The azimuth of the pendulum (423 20i 25 ), 

422-423 

736-752. The Motion of a Free 

Point 424-433 

736. The acceleration of a free point in 

any direction (424 6 ), . . . 424 

737. The rotation-area with reference to 

the moment of the force about an 
axis (424.2,,), .... 424-425 

738. The potential for a central force pro- 



portional to the distance. The 
path is a conic section (425i 2 ), . 425 

739. The area for forces directed towards 

a line is proportional to the time, . 425 

740. The path for the case of forces di- 

rected towards a line investigated 
by means of the peculiar coordi- 
nates of the distances from two 
fixed points of the line (426 163 ,), 
(427 3 ), (428„_ M ), . . . 425-428 

741. The special cases of § 740 may be 

combined into one by addition, . 428 

742. Cases in which the forms of § 740 

are expressed by elliptic integrals 
(428 27 _ 29 ), .... 428-429 

743. Case of § 740, in which there are two 

forces which follow the law of 
gravitation, 429 

744. Case of § 740, in which there is 

one force proportional to the dis- 
tance, 429 

745. Case of § 740, in which there is one 

force inversely proportional to the 
distance from the fixed line, . 429-430 

746. Restriction of the law of force for 

motion upon a given curve, . . 430 

747. Bonnet's theorem for combination 

of forces which produce a given 
motion (430 30 ), .... 430 

748. General value of the potential for 

§ 746 (481,), 431 

749. Cases in which the curve of § 746 is 

a parabola (4 31 17 _ 1922 _ 24 go^!), . .431 

750. Case in which the curve of § 746 is a 

conic section (432 7 _ 12 ), . . . 432 

751. Case in which the curve of § 746 is a 

cycloid (432 23 _ 25 ), . . . .432 

752. Case in which the curve of § 746 is a 

circle, or in which the surface of 
free motion is a sphere, . . 432-433 



CHAPTER XII. 

motion of rotation. 

753. Rotation-area defined. Principle of I 754. The parallelopiped of rotation- 

the conservation of areas, . 433-434] areas, ...... 434 



ANALYTICAL TABLE OF CONTENTS. 



XXXV11 



755-797. Rotation of a Solid Body, 

434-458 

755. The moments of inertia and the in- 

verse ellipsoid of inertia, . 434-436 

756. Rotation about a principal axis pro- 

duces no rotation-area about the 
other principal axes, . . . 436 

757. The plane of maximum rotation-area 

is conjugate to the axis of rotation, 436 

758. The ellipsoid of inertia, . . .436 

759. Position of the axis of maximum ro- 

tation-area with reference to the 
axis of rotation in the direct and 
inverse ellipsoids of inertia, . 436-43 7 

760. Euleb's equations for the rotation of 

a solid (43722,30), . . . .437 

761. The equation of living forces in the 

rotation of a solid (438 3 , ), . 437-438 

762-769. Rotation of a Solid Body 
which is subject to no ex- 
TERNAL Action, . . . 438-443 

762. The velocity of rotation of this solid 

is proportional to the correspond- 
ing diameter of the inverse ellip- 
soid (4383,), 438 

763. The velocity of rotation about the 

axis of maximum rotation-area and 
the distance of the tangent plane 
at the extremity of the axis of rota- 
tion are invariable. Poinsot's 
mode of conceiving the rotation 
(439,), 438-439 

764. Permanency of the instantaneous axis 

and of the axis of maximum rota- 
tion-area in the body (439 23 ), (440 6 ), 

439-440 

765. Surfaces of the instantaneous axes 

in space (442.,), . . . 440-442 

766. The velocity of the instantaneous 

axis in the body (442 10 ), . . 442 

767. Case in which the axis of maximum 

rotation-area describes the circular 
section ; corresponding spiral path 
of the axis of rotation (442 1927 _ 31 ), 

442-443 

768. Case in which the ellipsoids of iner- 

tia are surfaces of revolution, . 443 



769. The analysis of this case may be ex- 

tended to that of a solid rotating 
about a fixed point without the ac- 
tion of external forces, . . . 443 

770-783. The Gyroscope and the 

Top, 443-451 

770. Motion of a solid of revolution about 

a fixed point (444 3 ), . . 443-444 

771. The rotation about the axis of revo- 

lution is uniform, .... 444 

772. The motion of the gyroscope (444 31 ), 

(445 ; _ 18 ), .... 444-445 

773. The motion of the gyroscope ex- 

pressed by elliptic integrals 
(446,. w ), .... 445-446 

774. When the velocity of rotation van- 

ishes, the gyroscope is a spherical 
pendulum, .. . . . . 446 

775. Case in which the gyroscope de- 

scribes a horizontal circle, . .446 

776. Major Barnard's case of the gyro- 

scope in which the initial velocity 

of the axis vanishes, . . . 447 

777. Case in which the azimuthal motion 

of the axis is reversed during the 
oscillation, . . . . .447 

778. Case in which the axis of the gyro- 

scope becomes the downward ver- 
tical during the oscillation (448 6 ), 

447-448 

779. Case in which the axis of the gyro- 

scope becomes the upward vertical 
during the oscillation (448 18 ), . 448 

780. Case in which the velocity of the axis 

vanishes for the upward vertical, . 448 

781. Case in which the axis constantly 

approaches the upward vertical 
without reaching it (44 9g_ 16 ) , . .449 

782. The theory of the top (444,,), (445,), 

(449 28 ), 449-450 

783. Friction in the case of the gyroscope. 

The sleeping of the top, . 450-451 

784-791. The Devil on Two Sticks 

and the Child's Hoop, . 451-456 

784. Theory of the motion of the devil 

on two sticks (452 8|12 ), . . 451-452 



XXXV111 



ANALYTICAL TABLE OF CONTENTS. 



785. The axis of the devil cannot be- 

come vertical in the general case 
(452 M ), 452 

786. Case of the devil in which there is 

no rotation-area about the vertical 
axis, and in which the axis of the 
devil may become horizontal, 452-453 

787. Case of the devil in which there is 

no rotation-area about the vertical 
axis, and in which the axis of the 
devil cannot become horizontal, . 453 

788. Case in which the axis of the devil 

may become horizontal with a gyra- 
tion about the vertical axis ; and 
the corresponding case when it 
cannot become horizontal, . 453-454 

789. Case in which the axis of the devil 

may become vertical, . . . 454 

790. Theory of the body rolling upon a 

horizontal plane (455 20 2i), • 454-455 



791. Peculiar motion of the hoop when it 

is nearly falling, . . . 455-456 

792-794. Rotary Progression, Nuta- 
tion, and Variation, . 456-457 

792. Definition of nutation, progression, 

and variation of axes, . . . 456 

793. Accelerative forces which produce 

nutation, progression, or variation, 456 

794. Cases of these various actions, . . 457 

795-797. Rolling and Sliding Mo- 
tion, 457-458 

795. General theory of rolling motion 

(457 27 ), 457 

796. General theory of sliding motion 

(458 5 ), . 457-458 

797. Theory of sliding with friction ; case 

in which the sliding disappears, and 
the motion becomes that of rolling, 458 



CHAPTER XIII. 

MOTION OF SYSTEMS. 



798. Principles of power, translation and 

rotation applicable to all systems, . 458 

799. Forces of different orders, disturb- 

ing forces and perturbations, . 458-459 

800. Division of the system into partial 

systems, 459 

801-805. Lagrange's Method of Per- 
turbations, . . . 459-462 

801. Method of the variation of the arbi- 

trary constants (460 20 ), . . 459-460 

802. Combination of divers modes of vari- 

ation (461 6 , 9 ), . • • 460-461 

803. Derivative of the disturbing force 

with reference to an arbitrary con- 
stant (461 23 ), (462 2 ), . . 461-462 

804. Special case in which the arbitrary 

constants are the initial values of 
the variables (462 n ,i 2 ), . . .462 

805. Variation of the constant of power 

(4622s), 462 



806-808. Laplace's Method of Per- 
turbations, . . . 462-465 

806. Direct integration of the disturbed 

functions which are equivalent to 
the undisturbed arbitrary con- 
stants, 462-463 

807. Special case of frequent occurrence 

in planetary perturbations, . 463-464 

808. Perturbations of a projectile, . . 464 

809-818. Hansen's Method of Per- 
turbations, . . . 465-469 

809. The first principle of this method, and 

the expression of the time as an in- 
variable arbitrary constant (465 8 ), 465 

810. The principle of § 809, applied to the 

case of §808 (465i8_2o), . . .465 

811. The principle of § 809, applied to the 

case of §807, and especially when 
the disturbing force has a simple, 
periodic form (465 25 ), (466 2j 8 ), 465-466 



ANALYTICAL TABLE OF CONTENTS. 



XXXIX 



812. The second principle of this method 

or the application of the perturba- 
tions to the element of time, so that 
one of the functions may involve 
no other element of perturbation 
(466 25 ), 466-467 

813. Additional perturbation in the case 

of §812 of any other function 
(467 10 ), 467 

814. Other forms of the perturbation in 

the first approximation (467 22 ), . 467 

815. Case in which the function of §812 

does not involve the velocities, 

467-468 

816. 817. Case in which the initial values 

of the functions of §§ 812 and 813 
are simply related to the arbitrary 
constants, ..... 468 

818. The further development of the 

methods of perturbation is reserved 
for celestial mechanics, . . 468-469 

819-824. Small Oscillations, . 469-472 

819. The theory of small oscillations is re- 

duced to the integration of a sys- 
tem of linear differential equations 
(469*), 469 

820. The superposition of small oscilla- 

tions, 469-470 

821. Integration of the equations of § 820 

(4 70 17 . 24 ), ' 470 



822. 



823. 



824. 



Admissible forms of small oscillations 
correspond to stable elements of 
equilibrium (470 28 ), . . 470-471 

Independent elements of oscillation 
(471 w ), 471-472 

Oscillation and vibration pervade 
the phenomena of nature, . .472 



826. 



827. 



825-830. A System moving in a Re- 
sisting Medium, . . 472-475 
825. Equation for the determination of 
the Jacobian multiplier in such a 
system (472 31 ), . . .472-473 
The factoi's of the multiplier corre- 
spond to different laws of resistance, 473 
Cases in which the multiplier is 
unity, 474 

828. The multiplier when the resistance is 

proportional to the velocity, . 473-474 

829. The multiplier when the resistance 

is proportional to the square of the 
velocity, 474 

830. Equation of power for a system mov- 

ing in a resisting medium (474 15j21 ), 474 

831. Motion of the centre of gravity in a 

resisting medium (474 273 i), (475 3 ), 

474-475 

832. The rotation-area in a resisting me- 

dium (475 10 _ 23 ), . . . .475 



833. The Conclusion, 



476-477 



APPENDIX. 

Note A. On the Force of Moving I Note B. On the Theory of Ortho- 
Bodies, .... 479-480 1 graphic Projections, . . 481 



List of Errata, 



483-486 1 Alphabetical Index, . 



487-496 



ANALYTIC MECHANICS. 



CHAPTER I. 

MOTION, FORCE, AND MATTEE. 



§ 1. Motion is an essential element of all physical phenomena ; 
and its introduction into the universe of matter was necessarily the 
preliminary act of creation. The earth must have remained forever 
" without form, and void ; " and eternal darkness must have been 
upon the face of the deep, if the Spirit of God had not first "moved 
upon the face of the waters." 

2. Motion appears to be the simplest manifestation of power, 
and the idea of force seems to be primitively derived from the 
conscious effort which is required to produce motion. Force may, 
then, be regarded as having a spiritual origin, and when it is 
imparted to the physical world, motion is its usual form of mechan- 
ical exhibition. 

3. Matter is purely inert. It is susceptible of receiving and 
containing any amount of mechanical force which may be commu- 
nicated to it, but cannot originate new force or, in any way, trans- 
form the force which it has received. 

1 



— 2 



CHAPTER II. 

MEASURE OF MOTION AND FORCE. 



MEASUKE OP MOTION. 

§ 4. Uniform Motion is that of a body which describes equal 
spaces in equal times. 

5. Velocity is the measure of motion. In the case of uniform 
motion it is the distance passed over in a given time, which is 
assumed as the unit of time, and, in any case, it is at each instant 
the space which the body would pass over, if it preserved the same 
motion during a unit of time. 

6. If the space described by a body in the time t is denoted 
by s, the expression for the velocity v is, in the case of uniform 
motion, 

s 

If the differential is denoted by d and the derivative by D, the 
expression for the velocity is, in any case, 

ds t-> 



II. 

MEASURE OP FORCE. 

7. Experiments have shown that the exertion which is re- 
quired to move any body, is proportional to the product of the 



— 3 — 

intensity of the effort into the space through which it is exerted. 
This product is, then, the proper measure of the whole amount 
of force which is necessary to the production of the motion ; 
long established custom has, however, limited the use of the 
word force to designate the intensity of the effort, and the ivhole 
amount of exertion may be denoted by the term poivcr. Hence, if 
the power P is produced by the exertion of a constant force F, 
acting through the space s, the expression of the force is 

P 



F= 



s 



But if the force is variable in its action, the expression of its 
intensity at any point is 

F=~ = D S F. 

ds 

8. It is found by observation that the force of a moving body 
is proportional to its velocity. Thus, if m is the force of a body 
when it moves with the unit of velocity, its force, when it has 
the velocity v, is mv. 

9. Different bodies have different intensities of force when 
they move with the same velocity. The mass of a body is its 
force, when it moves with the unit of velocity ; thus, in in the 
preceding article, denotes the mass of the body. 

10. The force communicated to a freely moving body, by a 
force which acts in the direction of the motion, is found to be the 
product of the intensity of the acting force, multiplied by the 
time of its action. Thus, if the mass m, acted upon by the con- 
stant force F, for the time t, in the direction of its motion, has 
its velocity increased by v, the addition to the force of the mov- 
ing body is 

mv = Ft. 



In case the acting force is not constant, the rate at which the 
force of the body increases is 

mJD t v = F. 
III. 

FOKCE OF MOVING BODIES. 

11. The power with ivhich a body moves is equal to the product of 
one half of its mass multiplied by the square of its velocity. 

For if the body, of which the mass is m, is acted upon by 
the force F, until from the state of rest it reaches the velocity 
v, the power P, which has been communicated to it, and which it 
consequently retains, must, by (3 14 ) * and (4 3 ), give the equation 

D s P = mD t v. 
The derivative of P relatively to t, is by (2 24 ) 

D t P = D S P. D t s = vD s P = mvD t v. 
The integral of this equation is 

P = imv 2 , 

to which no constant is to be added, because the power vanishes 
with the velocity. [Note A.) 

12. Hence the power of a moving body is equal to one half 
of the product of its force multiplied by its velocity. 

* The form of reference here given is by means of numbers, of which the leading 
number refers to the page, and the secondary number, which is printed in smaller 
type, refers to the place upon the page, estimated from the top of the page, in lines of 
equal typographic interval. Printed marks, corresponding to these intervals, accom- 
pany each copy of the work. Thus, (3i 4 ) denotes the equation which is at the 14th 
typographic interval from the top of the third page. 



13. It is convenient to refer the measure of force to the 
unit of muss as a standard. Thus, if F is the force exerted upon 
each unit of mass, the force exerted upon the body of which the 
mass is m, is mF. With the F, used in this sense, (4 3 ) becomes 

D,v = F. 



>♦- 



CHAPTER III. 

FUNDAMENTAL PRINCIPLES OF PEST AND MOTION. 



TENDENCY TO MOTION. 

§ 14. A system of moving bodies may be regarded mechanically as a 
system of forces or poivers, which must be the exact equivalent of all the 
forces or powers which, by simultaneous or successive communication to the 
bodies, are united in its formation. 

This results from the inertness of matter, and its incapacity to 
increase, diminish, or vary in any way, the power which it contains. 

15. It also follows from its inertness, that matter yields instan- 
taneously to every force, and cannot resist any tendency to the 
communication or abstraction of power. With a system which is 
at rest, there can consequently be no tendency to the communi- 
cation of power. 

16. The tendency of any body or system of bodies to move 
in any given way is easily ascertained. It is only necessary to sup- 
pose the system moved with the proposed motion to an infinitesimal 



— 6 — 

distance. The product of the corresponding distance, by which each 
body of the system advances in the direction in which each force 
acts, multiplied by the intensity of the force is, by § 7, the corre- 
sponding power which the force communicates directly to the 
body, and through it to the system. 

The ivhole amount of power which is thus communicated by all the 
forces to the system, or rather its ratio to the infinitesimal element of the 
proposed motion is evidently the measure of the tendency of the system to 
this proposed motion. 

It must be observed that, when a body moves in a direction 
opposite to that of the action of the force, the corresponding product 
is negative, and must be used with the negative sign in forming the 
algebraical sum, which represents the whole amount of power com- 
municated to the system. 

17. By a skilful use of the principles of the preceding sec- 
tion, all the elementary tendencies to motion in a system may be 
determined, and, therefore, all the elements of change of motion in 
the system which is actually moving, or all the conditions of equi- 
librium in the system which is at rest. Thus, let 

mi, m 2 , m 3 , &c, denote the masses of a system of bodies; 

F lf F[, F", &c, the forces which act upon each unit of m± ; 

F 2 , F 2 , F 2 , &c, the forces which act upon each unit of m 2 ; 

&c. &c. ; 

df x , df[, df{, &c, the distances by which m x advances in the 

direction of the forces F x , F[, F'{, &c, in consequence of 

any proposed motion ; 
df 2 , df 2 , df 2 , &c. ; tT/ 3 , &c, the corresponding distances for the 

other bodies and forces of the system ; 
-2" ', the sum of all quantities of the same kind, obtained by 

changing the accents ; 



— 7 — 

2 1} the sum of all quantities of the same kind, obtained by 

changing the underwritten numbers ; 
JS^j the sum of all quantities of the same kind, obtained by 

all admissible combinations of both changes. 

The power communicated to the system by the proposed 
motion through m 1: m 2 , &c, is 

S'miFJfi = m, {F x df x + F[df[ + &c.) 
Z'm 2 F 2 df 2 = m 2 {F 2 df 2 + F' 2 df 2 + &c.) 
&c. &c. ; 

and the whole power communicated is 

= Z'm x F x df x + 2'm 2 F 2 df 2 -f &c. 

This is, therefore, the complete measure of the tendency in the 
system to the proposed motion, or of the change of motion which 
the moving system would experience in the direction of the pro- 
posed motion. But by a simple change in the values of cT/i, df[, 
cT/* 2 , df 2 , &c, the tendency to any other proposed motion may be 
measured ; and, in the same way, all the elements of the change of 
motion may be definitely ascertained. 

II. 

EQUATIONS OP MOTION AND REST. 

18. If, instead of the given forces, each body were acted upon 
by a force in the direction of its motion, and of such an intensity as 
to produce the exact change of velocity which it undergoes, this 
new system of forces would precisely correspond to that actually 
imparted to the moving bodies, and would be the exact equivalent 
of the given system of forces. Let 



— 8 — 

v i> v 2? ?, 3? & c - denote the velocities of the bodies; 

ds x , ds 2 , ds 3 , &c., the distances by which, in consequence of 
the proposed arbitrary motion of the preceding section, 
the bodies advance in the actual direction of this motion ; 

and then from (4 3 ) 

D t v x , D t Vz, D t v z , &c, are the intensities of the new forces 
relatively to the unit of mass. 

The whole power communicated by the new system of forces 
with the proposed motion becomes, then, 

2 1 m 1 D t v 1 ds l = nhDt^h -f- m 2 D t v 2 ds 2 -j- &c, 

and it must, therefore, be equal to the expression (7 13 ) of the 
power communicated by the given forces. Hence, 

S[ m x F x 8f x = 2 X m x D t i\ ds x , 

or by transposition 

Z 1 m l {D t v l ds 1 —Z'F 1 df 1 ) = o. 

When the system is at rest, this equation becomes 

19. The equation (8 18 ) in the case of motion, or the equation 
(8 20 ) in the case of rest, although it appears to be a single equation, 
involves in fact as many equations as there are distinct elements of 
motion or rest in the system of bodies. For every such element 
gives a different set of values of df l} df[, df 2 , &c, ds 1} ds 2 , &c, which, 
substituted in (8 18 ) or (8 20 ), produce a corresponding equation. 
These equations, therefore, involve all the necessary conditions of 
motion or rest in every mechanical problem. All that remains, 
then, is to determine, by geometrical analysis, the various elements 
of motion or rest, and to integrate and interpret the algebraical 



— 9 — 

equations, into which (8 18 ) and (8 20 ) are finally decomposed. The 
Mecanique Analytique of the ever-living Lagrange contains the general 
forms of investigation with unequalled elegance and perspicuity. 
But the special modes of analysis, which are peculiarly adapted to 
the illustration and development of particular problems, have been 
too much neglected, and the attention of }^outhful explorers is 
earnestly invited to this unbounded field of research. 



=>♦< 



CHAPTER IV. 

ELEMENTS OF MOTION. 



MOTION OF TRANSLATION. 

§ 20. A single material point may be moved to an infinitesimal 
distance in any direction, which may be defined by either of the 
methods known to geometers, by the reference, for instance, to the 
directions of three mutually perpendicular axes. By the known 
theory of projections, [Note B,) the distance by which the point 
advances in the direction of its actual motion, or in any other direc- 
tion, may be fully determined from the distances which it advances 
in these three directions. The three distances, moved in the direc- 
tions of the axes, which are simply the projections of the proposed 
motion upon the three axes, are the three independent elements of 
motion which completely define the elementary motion of the single point. 

2 



— 10 — 

Thus if 

dp denotes the proposed elementary motion, if 

P> p P> denote the angles which this motion makes with the 

three mutually perpendicular axes, called the axes of x, 
y, and g, and 
dx, dy, dz, the projections of dp upon the axes, 

the expressions for these projections are, 

dx = cos P . dp, 
dy — cos*, dp, 

ds = co&P .dp. 

If, in general, 

P denotes the angle which the directions of p and q make 
with each other, the distance by which the point 
advances, in consequence of the proposed motion, in 
the direction of/ is, by the theory of projections, 

df = cos P, . dp 

— cos * . dx -I- cos * . df/ -4- cos-' . dz 

x i y z 

= J£\. cos* .dx : 
in which 

JS X denotes the sum of all the similar terms obtained by pro- 
ceeding from one axis to each of the others. 

21. The most important of all the elementary motions of a 
system of bodies are those which, being independent of the peculiar 
constitution of the system, may be common to all systems. Such 



— 11 — 

motions must be possible, even if the bodies which compose the sys- 
tem, do not change their mutual positions, but are so rigidly fixed 
that the whole may be regarded as one solid body. It will be 
shown that there are but two distinct classes of such motions, 
namely, those of translation and those of rotation. 

22. The motion of translation is that by which all the points of 
a body, or system of bodies, are transported through the same dis- 
tance in the same direction. The projections of an elementary 
translation upon three rectangular axes are given by equations 
(10 10 _ n ), while (10 2 i), is the expression of the distance by which the 
system, or any one of its bodies, advances in any direction, such as 
that of /, by reason of the proposed translation. 

23. Any number of different elementary translations may be 
supposed to be given at the same time to a system, and the result- 
ing motion will be such an elementary translation, that its projec- 
tion, estimated in any direction, will be the sum of the projections 
of the elementary translations estimated in the same direction. 

Two coexistent elementary translations may be combined geo- 
metrically by setting off from any point two lines of the same 
length with the elementary motions, and in the same direction with 
them ; and if a parallelogram is described upon these two lines as 
sides, the diagonal, which is drawn from the given point, will rep- 
resent in distance and direction the resulting elementary transla- 
tion. 

In the same way the geometrical resultant of the combination 
of three elementary translations may be represented by the diago- 
nal of a parallelopiped described upon the lines which represent the 
component translations. But this parallelopiped vanishes when the 
three lines are in the same plane. 



— 12 — 
II. 

MOTION OF KOTATION. 

§ 24. The motion of rotation is that by which all the points of 
a body or system of bodies turn about a fixed line in the body, 
which line is called the axis of rotation. If one stands with his feet 
against the axes of rotation, and his body perpendicular to it, and 
faces in the direction of the rotation, the positive direction of the 
axis of rotation is, in this treatise, regarded as lying upon his right 
hand, and its negative direction upon his left hand. It will be found 
convenient to represent a rotation geometrically by a distance pro- 
portional to the elementary angle of rotation, set off upon the posi- 
tive direction of the axis of rotation from any point taken at pleas- 
ure in the axis. If 

d6 denotes the elementary angle of rotation, and r the distance 
of a point of the body from the axis of rotation ; 

rd& is the elementary distance through which the point moves 
in consequence of the rotation. 

The form in which the subject of rotation will be here pre- 
sented, is not greatly modified from that which it has finally 
assumed in Poinsot's admirable exposition of the " Theory of the 
Motation of Bodies" as it is printed in the additions to the Connais- 
sance des Temps for 1854. 

25. When a body rotates about an axis, it is, in consequence of this 
rotation, simultaneously rotating about any other axis which passes through 
the same point, with an angle of rotation ivhich is represented by the 'pro- 
jection upon this neiv axis of the line ivhich represents the original angle of 
rotation. 

For by the angle of rotation 6 about the axis A (fig. 1), the 



— 13 — 

point P of the axis OB, which is at the distance 

r = PM 

from the axis OA, is moved through the distance r6. Although 
every point of the axis OA is actually at rest, it has with respect to 
P, a relative motion, which is the negative of that of P. A rota- 
tion $' about the axis OB gives the point N of the axis OA, which 
is in the plane drawn through P perpendicular to OB, and at the 
distance 

r' = PN 

from the axis of OB, a motion through the distance /&' taken nega- 
tively. This rotation is, then, the same with that which the actual 
rotation produces about the axis OB, if 

or t = t j= cos MPN 

a r 

= cos A OB; 

that is, if &' is equal to the projection of 6 upon OB. 

26. Three simultaneous elementary rotations about three axes, ivhich 
pass through the same point, and are not in the same plane, are equivalent 
to a single rotation about the diagonal of a parallelopiped, of ivhich the three 
lines representing the rotations are the sides, and the length of the diagonal 
represents the angle of elementary rotation. 

For the algebraic sum of the projections of the sides of the 
parallelopiped upon any line perpendicular to its diagonal is zero, 
and, therefore, there is no rotation about any such line. Hence the 
diagonal is stationary, that is, it is the axis of rotation. The whole 
amount of rotation, being the sum of the partial rotations about the 
diagonal which arise from the several rotations about the sides, is 
represented by the sum of the projections of the sides upon the 



— 14 — 

diagonal, which is, by the theory of projections, equal to the diago- 
nal itself. 

27. In the same way, two simultaneous rotations about the 
sides of a parallelogram may be combined into a single rotation 
about the diagonal. In short, simultaneous elementary rotations about 
axes which cat each other may be combined in the same way as elementary 
translations. 

28. To investigate the distance by which a given rotation 
causes any point of a body or system to advance in a given direc- 
tion, as that of / ; let 

d& be the elementary angle of rotation about the axis of p and 
/ the perpendicular let fall from the point upon the axis 
of rotation. 

Let a line be drawn through the given point, parallel to the 
projection of/ upon a plane, which is perpendicular to the axis of 
rotation, and let 

q be the perpendicular let fall upon this line from the point in 

which / meets the axis of rotation ; and 
~ the angle which / makes with the direction in which the 
point is moved by the* elementary rotation. 

The distance by which the point advances in the direction 
of /is 

d/=/cos %M = /cos e, sin £.<M 

= qsmP.dd, 

in which o should be taken positively when the point is moved 
towards the positive direction of /. 

29. If three rectangular axes are drawn through any point 
of the axis of rotation, and if 



— 15 — 

d6 x , d& y , d6 z are the projections of d& upon these axes, the dis- 
tance by which the point (x, y, z) is moved in the direc- 
tion of the axis of x, is 

dx = yd& z — sdd y 

= (?/cos^ — zcos p ) d& 

W Z y) 

= (cos r cos P — cos r cos P) rdd 
y z z y 

= (cos r cos P — cos r cos p ) cosec r . rd$ 

- y z z y ' p 

= (cos r cos-^ — cos r cos^) rd&. 
y z z y 

X 

There are similar expressions for the distances by which the 
point advances in the directions of the axes of y and z, which may 
be found by advancing each of the letters x, y, z, and x to the fol- 
loAving letter of the series. 

30. The two last members of equation (15 5 ) divided by r'dd 
give the following theorem ; 

cos d = cos r cos P — cos r cos^, 

x y z z y' 

in which the direction of & is that of the perpendicular to the com- 
mon plane of / and p, and it is taken upon that side of the plane 
for which, a positive rotation about it, would correspond to a 
motion through the acute angle from / to p. 

31. If there were another system of rectangular axes, x f , y', 
and s, equation (15 20 ) applied to them would give 

cos = cos u cos — cos y cos . 

x y z z y 

In this equation each of the letters x, y, z, and x might be 
advanced to the subsequent letter of the series, as well as each letter 



— 1G — 

of the series x', y r , z' , and x' . In this way eight other equations 
might be found similar to equation (15 28 ). 



III. 

COMBINED MOTIONS OF ROTATION AND TRANSLATION. 

32. An elementary rotation, combined with an elementary translation 
in any direction, ivhich is perpendicular to the axis of rotation, is equivalent 
to an equal elementary rotation about an axis ivhich is parallel to the origi- 
nal axis of rotation. The position of the new axis is determined by the con- 
dition that each of its points is carried by the original elementary rotation as 
far as by the elementary translation, but in an opposite direction. 

For the given motions cancel each other's action upon each 
point of the new axis, and leave it stationary ; while the original 
axis advances with the elementary translation by the exact dis- 
tance which corresponds to the elementary rotation about the new 
axis. The common plane of the two axes is perpendicular to the 
direction of the translation. 

33. Any simultaneous elementary rotations about axes parallel to each 
other are equivalent to a single rotation, equal to their sum, and about an axis 
parallel to the given axes, combined with an elementary translation equal to 
the motion ivhich any point of the new axis receives from their simultaneous 
action. 

This is a simple deduction from the preceding proposition. 

34. Let there be three rectangular axes, such that the new 
axis of rotation may be that of z ; let 

x \ilf\) x i> Uii & C -? be the points in which the original axes cut 

the plane of xy ; and let 
d& u d& 2 , &c, be the elementary angles of rotation about these 

axes. 



— 17 — 

The elementary rotation about the axis of z is 

The elementary translations in the directions of the axes of x 
and y are by (12 19 ) 

dy = — 2± x x dd 1 . 

The distances through which any point (x, y, z) is carried for- 
ward in the directions of the axes, are 

dx = dx —yd A = 2 1 y 1 d& 1 — y 2 1 d& 1 , 
dy = dy -\-z$& = — 2 1 x 1 d6 1 -\-x2 1 d6 1 . 

The points are, therefore, at rest for which 

= dz —ydG = 2 1 y 1 d$ 1 —y2 1 d6 1 , 

= Sy -j- xdd =—2 t x,d6 x -f- z2 1 d$ 1 . 

These are, therefore, the equations of the axis of rotation, an elementary 
rotation about which, equal to the sum of all the elementary rotations, is 
equivalent to the combination of all the elementary rotations. 

35. If the original elementary rotations are all equal, and if 
there are n axes of rotation, the equations (17 2 ) and (17n) become 

d6==nd6 lt 

dx=(2 1 y 1 — ny) d^, 
dy = (— 2 j x x -\- n x) d 6 X . 

The equations (17i 6 ) give for the single axis of rotation 

y=— > • 

n 

36. If any of these rotations are about an axis lying in the 
opposite to the assumed direction, they may be regarded as nega- 

3 



— 18 — 

tive rotations about axes having the same direction as the assumed 
one, and may be combined algebraically in the preceding sums. 

37. When the second member of equation (17 2 ) vanishes, the 
resulting rotation disappears, and the given elementary rotations 
are equivalent to the elementary translation defined by equa- 
tions (17 6 ). 

38. Two equal rotations about axes, which are parallel, but 
have opposite directions, constitute a combination which Poinsot 
has called a couple of rotations. 

A couple of elementary rotations is, therefore, equal to an elementary 
translation in a direction perpendicular to the common plane of the axes, 
and equal to the product of the distance between the axes multiplied by the 
elementary angle of rotation. 

39. Any simultaneous elementary motions of rotation and translation 
are equivalent to a single elementary rotation about an axis, combined ivith 
an elementary translation in the direction of the axis of rotation. 

For each rotation may be resolved into a translation and a 
rotation about an axis passing through any assumed point. But all 
the elementary rotations about axes passing through the same point 
are equivalent to a single rotation about an axis passing through 
the point, and all the translations are equivalent to a single transla- 
tion. The single translation may be resolved into two translations, 
of which one is parallel, and the other perpendicular to the single 
axis of rotation. The translation, which is perpendicular to the 
axis of rotation, combined with the rotation, is equivalent to a sin- 
gle rotation about an axis, parallel to the single axis, and, therefore, 
having the same direction with the remaining translation. 

40. Every possible motion of a rigid system or body is equivalent to 
a combination of the motions of translation and rotation. 

This is evident, if it can be shown that, by such a combination 
of motions, any three points, A, B, and C, of the system, can be car- 



— 19 — 

ried to any positions, A, B', and C, in which it is possible for them 
to be placed. For three points of a rigid system not in the same 
straight line completely determine, by their position, that of the 
whole system. Now, by a translation of the system, equal to that 
by which A might be directly moved from A to A r , the point A is 
actually brought to the position A. By a subsequent motion of 
rotation about an axis, which is "perpendicular to each of the lines 
AB and A B', the point B may be moved to B r ; and then by a 
rotation about AB' the point O may be carried to C. Hence the 
whole motion is accomplished by one translation and two rotations. 
Every elementary motion of a rigid system must then be 
equivalent to a single rotation about an axis and a translation in 
the direction of the axis of rotation. This motion is perfectly rep- 
resented by that of the screw, whose helix causes it to advance in 
the direction of the axis about which it is turning. 

41. During each instant of its motion, a rigid system rotates 
about an axis, which is called the instantaneous axis of rotation. This 
axis is generally varying its position in the system and in space 
from one instant to another, which renders it difficult to form 
a distinct conception of the nature of the corresponding motion of 
the system. 

42. In attempting to conceive of the motion of a rigid system, 
it is expedient, at first, to neglect the translation in the direction of 
the axis of rotation, and to assume that the motion is solely that 
of rotation. The successive positions of the axis of rotation in the 
system form by their union a surface which tarns with the system; 
and its successive positions in space form another fixed surface. In 
the motion now considered, the moving surface rolls on the fixed 
surface without sliding, and carries the system with it. 

43. If the axis of rotation does not move perpendicularly to 
itself each of these surfaces is evidently a developable surface, and 



— 20 — 

in the act of rolling the line of retrogression of the one falls upon 
that of the other; so that these two lines are of the same length. 
Upon the surfaces, developed into a plane, the two lines of retro- 
gression will be precisely alike. 

In combining with this rotation the translation in the direction 
of the axis of rotation, the surface, generated bj the instantaneous 
axis in the moving system, remains unchanged. But the fixed sur- 
face, generated by the instantaneous axis, is changed ; it is still a 
developable surface obtained from that in which the translation is 
neglected, by adding to each element of the arc of the curve of 
retrogression, the elementary translation in the direction of the axis 
of rotation. In the actual motion, the moving surface rolls upon 
the fixed surface, and glides simultaneously in the direction of the 
line of contact, so as to keep the curves of retrogression constantly 
in contact. 

In this general case, the whole length of the arc of the fixed 
curve of retrogression is equal to that of the moving curve aug- 
mented by the whole amount of translation in the direction of the 
axis of rotation. 

When the elementary translation is equal to the elementary 
arc of the moving curve of retrogression, but lies in the opposite 
direction, there is a corresponding cusp in the fixed curve of retro- 
gression. 

A point of inflection in the curves of retrogression generally cor- 
responds to a change in the direction of the rotation. A similar 
combination of the translation with the rotation can be introduced 
into the general case of motion. 

44. When either of the surfaces of the instantaneous axis is 
a cone, the curve of retrogression is reduced to a point which is the 
vertex of the cone. When both of the surfaces are cones, there is no 
translation in the direction of the axis. 



— 21 — 

When either of the surfaces is a cylinder, both surfaces must 
be cylinders; and the lines of retrogression, removing to an infinite 
distance, cannot be used for guiding the motion of translation. 
But in this case, a section may be made of one of the cylinders per- 
pendicular to its axis, and in the actual motion the moving cylinder 
will move so as to keep the point, in which the perimeter of this 
section touches the other cylinder, upon a curve properly drawn 
upon that cylinder. 

45. The general motion of a rigid system may be conceived as 
a translation, equal to that of any one of its points assumed at will, 
combined with a rotation about an instantaneous axis of rotation 
passing through the point. If the translation is neglected, the rota- 
tion is effected as in § 42 by rolling a cone, of which the assumed 
point is the vertex, and which carries the system with it, in its 
motion, about a fixed cone, of which the same point is the vertex. 
The translation may be simultaneously effected by moving the two 
cones in space, with a translation equal to that which belongs to 
their vertex in the actual motion of the system. 

46. For all the points of the instantaneous axis in each of its 
positions, the corresponding centres of greatest curvature of either 
of the conical surfaces which it describes, are all upon the same 
straight line passing through the vertex. 

In the case of the right cone, or of the right cylinder, the axis 
of revolution is the line of the centres of greatest curvature. In all 
these investigations the plane may be regarded either as a cylinder 
of infinite radius, or as a cone, of which the angle at the vertex is 
equal to two right angles. 

47. The elementary rotation of the system may be conceived 
as decomposed into two elementary rotations about the lines of the 
centres of greatest curvature as axes of rotation. By the rotation 
about the line, which unites the centres of the fixed surface, the 



— 22 — 

instantaneous axis receives its elementary motion in space, and is 
carried to its proper position upon the fixed surface. By the rota- 
tion about the line which unites the centres of the moving surface, 
the system receives that additional rotation which is required to 
turn the moving surface into that position in which it may have the 
proper line of contact with the fixed surface. Each of these rota- 
tions produces a sliding of the moving upon the fixed surface ; but 
as the sliding produced by the one is just equal and opposite to that 
produced by the other rotation, the two rotations cancel each 
other's action in this respect, and there is no sliding in the 
combined motion, but a simple rolling of one surface upon the 
other. 

48. Let 

a f be the acute angle which the instantaneous axis of rota- 
tion makes with the line of the centres of curvature 
of the fixed surface ; 

a m that which it makes with the line of the centres of cur- 
vature of the moving surface, this angle being positive 
when the two lines of the centres are on opposite 
sides of the instantaneous axis, and negative, when 
they are upon the same side ; 

d a* the elementary angle by which the instantaneous axis 
changes its direction ; 

d & f the elementary angle of rotation about the line of cen- 
tres of the fixed surface ; and 

d 6 m the elementary angle of rotation about the line of cen- 
tres of the moving surface. 

Since the instantaneous axis must be carried forward by the 
rotation about the fixed axis, and backward by the rotation about 



the moving axis just as far as its actual change of position, its ele- 
mentary angle of change of direction is 

d w = d 6 f . sin a f = d & m . sin a m . 

But the combination of the two rotations about these axes 
gives the actual rotation about the instantaneous axis, and there- 
fore, 

d 6 = d 6 f . cos cc/-\- d Q m . cos a m 

= (cotay-j-cotoj^) da> 

_ sin Ov+O ftrc 
sin «y sin a m 

49. When the surfaces described by the instantaneous axis are 
cylinders, let 

(jy and Q m be the respective radii of greatest curvature of the 
fixed and moving surfaces at any point of their mutual 
contact ; and 

dp the elementary distance which the instantaneous axis moves 
in a direction perpendicular to itself. 

The conditions of the motion of the instantaneous axis give the 
equations 

in which the upper sign corresponds to the case where the lines of 
the centres of curvature are upon opposite sides of the instanta- 
neous axis, and the lower sign to that in which they are upon the 
same side. The rotation about the instantaneous axis is 



d6 = d& f +d6 m 



— 24 — 
IV. 

SPECIAL ELEMENTS OF MOTION AND EQUATIONS OF CONDITION. 

50. The variation of each independent element of position of 
a system gives an independent element of motion. Bnt the ele- 
ments of position are various, and must be selected in each case 
with special reference to the problem under discussion. It often 
occurs that parts of the system are rigidly connected ; such parts 
are themselves rigid systems, and subject only to motions of trans- 
lation and rotation, and, therefore, none but such elements are 
required for the investigation of their motions. 

Points of the system are sometimes restrained to move upon 
given surfaces, and, in this case, it may be expedient to introduce 
elements' of position dependent upon the principal lines of curva- 
ture of these surfaces, or elements, in reference to which the sur- 
faces are peculiarly simple or symmetrical. Points of the system 
may be compelled to preserve simple geometrical relations to each 
other, which may suggest appropriate elements of position to the 
skilful analyst; or he may find indications to direct his choice in 
the very nature of the motion itself. 

51. It is often desirable to adopt a combination of elements 
of position which are not wholly independent of each other, but are 
subject to certain mutual restrictions. These restrictions, when 
they are expressed algebraically, are called equations of condition. 
They may assume the differential form of equations between the 
elementary motions ; or they may be finite equations between the 
elements of position, in which case they may be reduced by differ- 
entiation to equations between the elementary motions. 

By means of the equations of condition, as many of the ele- 
ments of motion may be determined in terms of the rest as there 



— 25 — 

are equations of condition ; and the remaining elementary motions 
may be regarded as independent of each other. 

52. Instead of introducing into the equations (8 18 ) and (8 20 ) of 
motion and rest the special values of ds ly ds 2 , &c, df ly df 2 , &c, for 
each particular element of motion, their general values may be 
found in terms of all these elements. When the elementary 
motions are wholly independent, their coefficients in these equa- 
tions give, when they are equalled to zero, the same equations 
which would have been obtained by the special investigations. 
But when the elements are not independent, all, except the inde- 
pendent elements can be eliminated by means of the values given 
by the equations of condition. 

The equations (8 1S ) and (8 20 ) of motion and rest, on account of 
their differential form, are necessarily linear in reference to the ele- 
mentary motions ; and the differential equations of condition are 
likewise linear. The proposed elimination may therefore be con- 
ducted by the method of multipliers. By this process each differential 
equation, multiplied by an unknown quantity, is to be added to the 
given equation of motion or rest. The unknown multipliers are to 
be determined by the conditions that the coefficients of the elemen- 
tary motions, which are to be eliminated, become equal to zero. 
Since the remaining elementary motions are independent of each 
other, their coefficients must also be equalled to zero. In the sum, 
therefore, obtained by the addition of the equations, each of the 
coefficients of the elementary motions is equal to zero. The num- 
ber of unknown quantities is increased in this process by that of the 
unknown multipliers ; but, because there are as many equations of 
condition as there are multipliers, the whole number of equations, 
including the equations of condition, in their finite form, is just 
sufficient to determine the values of the multipliers and of all the 
elements of position. 

4 



— 2G — 

53. Let 

be one of the equations of condition in its finite form ; and let its 
differential form be 

dL x = 0. 
Let also, 

I be the unknown multiplier by which it is to be multiplied. 

The sum obtained by adding the similar products of all the equa- 
tions of condition to equation (8 18 ) or (8 20 ) is 

2[m 1 F 1 df 1 -\-2 1 l 1 dX 1 = Q, 

which is the equation of motion or rest, and in which the general 
values of ds l7 df x , &c, are to be substituted, and the coefficient of 
each elementary motion is to be equalled to zero. 

54. Each equation of condition becomes the equation of a 
surface, to which any one of the points whose elements of position 
occur in the equation is restricted, provided that, for the moment, 
the variations of all the other elements are neglected. Since the 
point is restricted to move upon the surface, it cannot move in the 
direction of the normal to the surface. Let a system of three rec- 
tangular axes be adopted, and let 

iVbe the normal to the surface. 

Its variation, arising from the variation of coordinates, which may 
be regarded as the elements of position of the point, is 

If the equation of the surface is (26 2 ), with the omission of the num- 



— 27 — 

bers written below, which may be neglected in the general discus- 
sion, its variation is 

dL = Z x D x Ux. 

Let, then, 

and the angle, made by the normal with one of the axes, is given 
by the equation 

X __D X L 

cos^—^-; 

which substituted in (26 29 ) gives 

Z x D x L8x 8L 



djsr-. 



M M 



Hence the equation of condition with its multiplier may be writ- 
ten in the form 

IdL = X3IdJY= ; 

and this form may be substituted in the equations (26 12 ) and (26 13 ) 
of motion and rest. 



— 28 



CHAPTER V. 

FORCES OF NATURE. 



I. 



EQUILIBRIUM, AND THE POSSIBILITY OF PERPETUAL MOTION. 

§ 55. It appears, at first sight, to be inconsistent with the 
assumed spiritual origin of force, that the principal forces of nature 
reside in centres of action, which are not thinking beings, but parti- 
cles of matter. The capacity of matter to receive force from mind 
in the form of motion, contain and exhibit it as motion, and commu- 
nicate it to other matter, under fixed laws, is not, however, less dim- 
cult or more conceivable than the capacity to receive and contain it 
in a more refined and latent form, from which it may become mani- 
fest under equally fixed laws. It is only, indeed, when force is thus 
separated from mind, and placed beyond the control of will, that it 
can be subject to precise laws, and admit of certain and reliable 
computation. 

56. The laws of the development of power in nature are of 
two classes. In the one class, the forces depend solely upon the 
relative positions of the bodies, and may be called fixed. In the 
other class, the forces depend, not only upon the positions of the 
bodies, but also upon their actual state of power, especially upon 
the velocities and directions of their motions ; and these forces may 
be called variable. 

57. The most fruitful and enlarged view of the fixed forces of 



— 29 — 

nature, and one which peculiarly corresponds to their laws of action 
so far as they have been observed, is to regard them as the mani- 
festations of the dynamic situation of the bodies which exhibit them. 
The dynamic situation depends solely upon the masses and posi- 
tions of the bodies; it is a condition of form, and its research is a 
problem of pure geometry. The algebraic function which embodies 
the idea of the dynamic state is called the potential. Its complete 
investigation and determination involves the solution of all the 
problems which can arise in regard to the power and the conditions 
of force of all systems, whether they are at rest or in motion, so far 
at least as the fixed forces of nature are concerned. 

The amount of power of a system is not to be inferred from its 
situation, although there is a certain measure of power appropriate 
to that situation. It is this latter power which is expressed by the 
potential of the system, and expressed as a function of all the ele- 
ments of position, by which the situation is defined. 

58. The power of a moving system increases or decreases with the 
power ivhich belongs to its situation, and the increase or decrease of its power 
is measured by that of its potential. 

59. Hence, if a system moves from a state of rest, its power is 
constantly equal to the excess of its potential over the initial value 
of the potential ; and it can never arrive at a position in which the 
potential would be less than its initial value. No system, indeed, 
can move to a situation in which the potential would be diminished 
more than the initial power of the system. 

60. When a system is in a permanent state of rest which the 
actual forces do not tend to disturb, its dynamic condition is such, 
that the power of the system is not changed by a slight change of 
position. Hence, 

The potential of a system ivhich is in equilibrium, is generally a maxi- 
mum or a minimum. The exceptional case of a condition of indiffer- 



— 30 — 

ence rarely occurs in nature ; but even this case may be philosophi- 
cally regarded as the combination of a maximum and minimum, or 
as the result of several such combinations. 

61. When a moving system passes through a position of equi- 
librium, or a position which is one of equilibrium in reference to 
the element of position with which the system is changing its place, 
the power of the system is either a maximum or a minimum, or in 
a condition of indifference. 

62. When a system, in a state of rest, is placed very near the 
position of equilibrium, it cannot tend to move away from the posi- 
tion of equilibrium, if the potential of that situation is a maximum 
relatively to the element by which the system is removed from 
it ; and it cannot tend to move towards the situation of equili- 
brium, if the potential is a minimum for the same element. On 
this account the equilibrium is stable, in reference to those elements 
for which the potential is a maximum, and it is unstable in reference 
to these elements, for which the potential is a minimum. 

63. As when a function changes in consequence of the change 
of any one of its variables, the maxima and minima succeed each 
other alternately ; in the motion of a system, the positions of stable 
and unstable equilibrium, relatively to the element of change of 
position, succeed each other alternately. Situations of equilibrium 
of indifference may be interposed without disturbing the order of 
succession of the situations of stable and unstable equilibrium. If 
the system returns to its initial position, it must have passed 
through an even number of such situations of equilibrium, rela- 
tively to the element of change of position, half of which must have 
been positions of stable, and the other half positions of unstable 
equilibrium. In general, these situations will not be positions of 
absolute equilibrium, but only such in reference to the changing 
element of motion. 



— 31 — 

64. Fixed forces might easily be imagined different from 
those of nature, and in the action of which the power of a moving 
system would depend upon its previous situations as well as upon 
its actual position. With such forces the increase or decrease of 
power of a system would vary with the path which it pursued in 
moving from one situation to another, and would be greater by one 
path than by another. The change of power for each element of 
any given path, would still be computed by the process of § 17, 
and thence the whole change of power would be obtained by inte- 
gration. If the motion of the system were reversed, and it were 
carried back through the same path to its initial position, its initial 
power would be restored. If, of two courses, by which a system 
could move from one situation to another, it were forced to go by 
that through which it would arrive, with the greater power at its 
final position, and if it were then made to return to its initial posi- 
tion by the other path, it would return with an increased power ; 
if it were again to move through the same circuit, it would again 
return with an equal additional increase of power ; and, by succes- 
sive repetitions of this process, the power might be increased to any, 
even to an infinite amount. Such a series of motions would receive 
the technical name of a perpetual motion, by which is to be under- 
stood, that of a system which would constantly return to the same 
position, Avith an increase of power, unless a portion of the power 
were drawn off in some way, and appropriated, if it were desired, to 
some species of work. A constitution of the fixed forces, such as 
that here supposed, and in which a perpetual motion would be pos- 
sible, may not, perhaps, be incompatible with the unbounded power 
of the Creator ; but, if it had been introduced into nature, it would 
have proved destructive to human belief, in the spiritual origin of 
force, and the necessity of a First Cause superior to matter, and 
would have subjected the grand plans of Divine benevolence to 
the will and caprice of man. 



— 32 — 

65. A surface, for each of whose points the potential has the 
same value, may be called a level surface. A level surface may be 
drawn through any point in space. 

Since the potential of every finite system of nature vanishes 
for an infinitely distant point, all the level surfaces of nature are finite, 
and, returning into themselves, include a space ivhich they wholly surround, 
with the exception of those level surfaces for which the potential is zero. 

66. A material point, placed upon a level surface, has no ten- 
dency to move in the direction of the surface, because there is no 
increase of power in such direction. The tendency of a material point 
to motion is, therefore, perpendicular to the level surface upon ivhich it is 
placed, 

67. If two level surfaces are drawn infinitely near to each 
other, a material point, placed upon either of them, tends to move in the 
direction, from the surface of the less potential toivards the other, ivith a 

force ivhich is measured by the quotient of the difference of the potentials of 
the two surfaces, divided by their distance apart. 

Hence, if the surfaces are, throughout, at the same distance 
apart, the disposition to motion is everywhere the same. 

If the surfaces were to intersect each other, the tendency to 
motion in the line of intersection would be infinite ; but, since there 
is no such infinite tendency to motion in nature, each level surface of 
nature must be wholly included within every other level surface, within which 
any portion of it is included. For the same reason, the potential in nature 
is always a continuous function. 

68. Within each level surface of nature there must be a point 
or points of maximum or minimum potential. A continuous 
curved line, drawn perpendicularly to each of the level surfaces 
which it intersects, represents a line of action or tendency to 
motion, and every such trajectory must finally terminate in one of 
the included points of maximum or minimum potential. Each of 



— 33 — 

these points may then be regarded as a centre of action, towards, or 
from which, all motion tends along the various trajectories, accord- 
ing as the point is that of a maximum or a minimum potential. 

69. If the potential has a constant value for any portion of space, 
this same constant value must extend throughout all thai space, including 
this portion, for which the potential and all its derivatives are finite and con- 
tinuous functions. For, in order that the potential may be absolutely 
constant for any finite extent, however small, all its derivatives 
must vanish. But it follows, from Taylor's Theorem, that the 
difference of the value of the potential for any portion of space, for 
which it is continuous and finite, as well as all its derivatives, is a 
linear function of its derivatives at any point of that space. The 
difference of the potential, therefore, vanishes, when all the deriva- 
tives vanish and the potential is constant. 

The portion of space, for which the derivatives are originally 
assumed to be constant, must be a solid, having the three dimen- 
sions of extension, in order that this theorem be applicable. 

70. Throughout any such portion of space, in which the 
potential is constant, there can be no tendency to motion in any 
direction. In such extent, therefore, there can be no mass of 
matter, for it is contrary to experience that there should be matter 
where there are no dynamical phenomena. 

71. In all the observed laws of material action, the potential, 
which belongs to the action of each particle of matter, is finite and 
continuous, as well as all its derivatives, for the whole extent of space 
exterior to the particle. Hence, the potential and its derivatives, 
for every system of nature, are finite and continuous functions 
throughout any portion of space which contains no material mass. 

72. Hence, it follows, that for every finite system of nature, any 
portion of space, in which the potential is constant, must be finite, and 
hounded on all sides by material masses. This portion of space cannot 

5 



— 34 — 

extend to infinity, because, if it were to have such an extent, the 
finite mass, which would be its inner limit, would exhibit no 
external indication of force ; whereas, it is obvious that no matter 
can ever have been observed, except by such a manifestation of its 
existence. 

73. There are forces in nature which are temjiorarily fixed, and 
for which the potential may vanish throughout all space exterior to 
the limit in which the centres of action are contained. 

74. The difference between the values of the potential for any 
two points may be computed by supposing a unit of mass to move 
from one point to the other upon any line taken at pleasure, and 
determining the change of power which it receives from this 
motion. The change of the potential may be computed for each 
force separately, and, in making the partial computations, it is 
sufficient to suppose the unit of mass to move from the level 
surface of one point to that of the other, and one of the perpen- 
dicular trajectories may be taken for the path of this motion. 

75. If, in any system, 

F, F', &c, are the forces ; 

/,/', &c, the directions in which they act ; and 
12 is the value of the potential ; 

the general expression of the potential for any point of the 

system is 

a = 2'fFdf i 

in which the limits of integration extend from the values of/,/', &c, 
which correspond to the position of the point, to infinity. The 
expression for the tendency to motion in any direction, as that of 
p, is 

D p Q = D p 2'fFdf. 



— oO — 

II. 

COMPOSITION AND RESOLUTION OF FORCES. 

76. No phenomenon is observed, in which a single force acts 
freely by itself. In all cases, various forces are combined ; and it 
is important, therefore, to ascertain what are the dynamical results 
of such combinations. 

77. A single force acts, at each point, perpendicularly to its 
level surface, with an intensity which is measured by the derivative 
of the potential, taken with reference to the element of direction of 
the force. The intensity of its action, in any other direction, is 
measured by the derivative, with reference to the element of that 
direction. If another level surface is drawn infinitely near the one 
which passes through the point, the action in any direction is 
inversely proportional to the length, intercepted by the surfaces, 
upon a straight line drawn in the given direction. But the surfaces 
may, for this purpose, be considered as reduced to their parallel 
tangent planes at the given point ; and the length, intercepted 
between two parallel planes, upon a straight line, is proportional to 
the secant of the angle which the line makes with the perpen- 
dicular to the plane. Hence, the action of a force in the direction 
of any line, is proportional to the cosine of the angle which it 
makes with the direction of the force. 

If, then, upon a straight line drawn in the direction of a force, 
a length is taken to represent the intensity of the force, the action 
in any direction is represented by the projection of this length 
upon that direction, or by using the word force for the representa- 
tive of the force, the proposition becomes, that the action of a force in 
any direction is the projection of the force upon that direction. 

78. When several forces act upon a point, their total action in 



— 3d — 

any direction is the algebraic sum of their projections upon that 
direction. 

79. When three forces, ivhich are not in the same plane, act upon a 
point, their combined action is equivalent to that of a single force, tvhich is 
represented in magnitude and direction by the diagonal of the parallelopiped 
constructed upon the three forces. 

For the algebraic sum of the projections of the forces upon any 
direction perpendicular to the diagonal, is zero, while that of the 
projections upon the diagonal is the diagonal itself. 

80. All the forces tvhich act upon a point, are equivalent to a single 
force, which is called their resultant. For a single point can only tend 
to move, with a certain intensity, in some one direction, however 
various may be the forces which act upon it ; and any such 
tendency to motion can be produced by one force acting upon 
the point. 

The actions of all the forces in three directions which are 
perpendicular to each other, can be found by § 78 ; and these three 
partial forces can then be combined by § 79 into one force which 
will be the resultant. But the following method of finding the 
resultant illustrates the use which may be made of the level 
surfaces. 

81. In considering the action of a force upon a fixed point in 
space, the variable character of the force for other points of space 
may be neglected, and its level surfaces may be regarded as parallel 
planes perpendicular to the direction of the force. Thus, it may be 
assumed that 

Ff is the potential of the force F, which acts in the direction 

of/; for 
D f {Ff) = F, is the intensity of the force ; and 

Ff =. a constant, or 
f= a constant, 



— 37 — 

is the equation of a plane perpendicular to /. Hence, the potential 
of all the forces which act upon the point, is 

If then 

P q is the resulting force resolved in the direction of q ; if 
p is the direction of the resultant, and 
P is the resultant ; 

the value of either of these forces is represented by the formula 

P q = D q a = 2'FDJ = 2'Fcoaf. 
But, by putting 

j?=js m p m ny=2 m pi, 

the condition that p is perpendicular to the level surface, for which 
the potential is constant, gives 









COS^ =z—f— = 

x L 


P x 
L' 




the 


value 


of the 


resultant is 










P 


= D p £2=2 x & 


&!>, 


, Ju 








= 2 X D X £2 cos?-. 


= Z X 


-* X 
1 








_ Z x PI & 








L L 










= L = sl(2 x P\ 


;)• 





82. By an elementary motion of translation, each point of a 
system is carried to the same distance in the same direction ; the 
potential of the system is changed, therefore, precisely as if all its 
points were united in one, and all the forces applied at this point. 
The tendency of a system to any motion of translation, is, then, the same as 



— 38 — 

that ivliich would arise from the action of a single force, equal to the 
resultant of all the forces, supposed to be applied at the same point. 

83. The moment of a force, ivith reference to a point, is the product 
of the force multiplied by its distance from the point. The moment 
of a force, ivith reference to a line, is the product of the projection of 
the force upon a plane perpendicular to the line multiplied by the 
distance of the force from the line. 

The moment of a force, with reference to a line, may be 
represented geometrically by a corresponding length taken upon 
the line, and the name of the moment may be given to its geomet- 
rical representative. 

The moment of a force, •with reference to a point, is the same 
with the moment, with reference to the line, which is drawn 
through the point perpendicular to the common plane of the point 
and the force. 

84. The moment of a force, ivith reference to a line passing through a 
point, is equal to the projection upon the line of the moment, with reference to 
the point. For the moment, with reference to the point, is equal to 
double the area of the triangle, of which the base is the force, and 
the altitude is the distance of the force from the point ; and the 
moment, with reference to the line, is equal to double the area of 
the triangle, of which the base is the projection of the force upon 
the plane perpendicular to the line, and the altitude is the distance 
of this projection from the line. But the latter of these triangles is 
the projection of the former upon the plane, and its area is equal to 
the product of the area of the former triangle, multiplied by the 
cosine of the angle of the planes of the two triangles. But the 
lines upon which the moments are represented, being respectively 
perpendicular to these planes, have the same mutual inclination. 
The moment, with reference to the line, is, therefore, equal to the 
product of the moment, with reference to the point, multiplied by 



— 39 — 

the cosine of the mutual angle of the moments ; that is, it is equal 
to the projection upon the line of the moment, with reference to 
the point. 

85. Hence it follows that the moments of forces, with refer- 
ence to points, may be combined by the same processes in which 
the forces themselves are combined, and that all the moments, with 
reference to a point, may he combined into one resultant moment. 

86. The tendency of the force F, of which the potential is 
Ff, to produce an elementary rotation, d&, about a line p, is 



But if 
(14 2G ) gives 



D (Ff) = FD d f. 
o is the distance of F from p, 



B f=QsmP; 
the projection of F upon the plane perpendicular to^;, being 

Fsm? 

the tendency to rotation about p becomes 

oJ^sin^ = the moment of F with reference to p ; 

that is, the moment of a force, with reference to a line, is the measure of its 
tendency to produce rotation about that line. 

87. The direction of the positive moment must be assumed to 
be the same with that of the axis, about which the tendency to 
rotation of the force is positive. 

88. The residtant moment of all the forces of a system, tvith reference 
to a point, is the measure of their tendency to produce rotation about that 
point. Hence, the one force, of which the moment is equal to the 
resultant moment, has the same tendency to produce rotation. 



— 40 — 

89. The resultant moment of all the forces which act upon a 
point, with reference to any line or to any other point, is the same 
with the moment of their resultant. For the point upon which the 
forces act tends to move in the direction of their resultant, with a 
force equal to its intensity, and its moment is, therefore, the 
measure of the tendency to motion. 

90. The moment of a force, with reference to a line p r , is 
equal to its moment, with reference to a parallel line p, increased 
by the moment of an equal and parallel force, acting at any point 
of the line p. For the distance of the original force from the line 
p r , is equal to its distance from the line p, increased by the distance 
froniji/ of the parallel force passing through p. 

91. Hence the resultant moment of any forces, with reference to a line 
p', is equal to their resultant moment, ivith reference to a parallel line p, 
increased by the moment, with reference to p ' , of equal and parallel forces 
acting at any point of the line p. 

92. The resultant moment of any forces, with reference to a point ' , 
is equal to their resultant, with reference to a point 0, increased by the 
moment, with reference to ' , of equal and parallel forces acting at 0. For 
this proposition is true for each pair of the parallel axes of two 
parallel systems of three rectangular axes, of which the points 
and 0' are the respective origins. 

93. A couple of forces is a system of two parallel and equal 
forces which act in different lines. 

94. The moment of a couple of forces has, for every point of space, 
the same value, which is equal to the moment of one of them for any point of 
the other. For two forces, equal and parallel to them, applied at any 
point, destroy each other's action, and their resultant vanishes. 

95. The tendency of a couple of forces to produce rotation 
about a point, is the same as that of any system of forces, when its 
moment is equal to the resultant moment of the system, with 



— 41 — 

reference to the point. But the couple has no tendency to 
produce a translation ; whereas the resultant of a system of equal 
and parallel forces, acting at the point, has all the tendency of the 
system to produce translation, but none to produce rotation about 
the point. Hence, the three forces, of which one is the resultant of the 
equal and parallel forces acting at a point, and the other tivo constitute a 
couple, of which the moment is the same with the resultant moment, with 
reference to the point, fully represent any system of forces in their tendency 
to produce rotation and translation. 

96. Since the position of the couple of forces is quite arbi- 
trary, one of the pair may be taken to act at the same point with 
the resultant of all the forces; and, by combining it with the 
resultant, the system of three forces may be reduced to two. 

97. A point can always be found in space, for which the 
moment of a given force has any assumed magnitude, and any 
direction which is perpendicular to the force. Because the distance 
of the point from the force, which is one of the factors of the 
moment, may vary from zero to infinity, and its direction from the 
force may be that of any perpendicular to the force. 

Hence, if the resultant moment, with reference to a point 0, 
of any system of forces, is decomposed into two moments, of which 
one has the same direction with the force, and the other is per- 
pendicular to it, another point 0' can be found, for which the 
moment of the resultant, acting at 0, is, in amount and direction, 
the negative of that component of the resultant moment for 0, 
which is perpendicular to the resultant. For the point 0', there- 
fore, the resultant moment, coincides in direction with the result- 
ant itself; and of the three corresponding forces which represent 
the tendency of the system to produce rotation and translation, the 
plane of the couple is perpendicular to the direction of the result- 
ant. 

6 



— 42 — 

98. If all the forces lie in the same plane, for any point of the 
plane the moment of each of the forces is perpendicular to the 
plane, and, therefore, the resultant moment is perpendicular to the 
plane. But the resultant of the parallel and equal forces acting at 
the point must, if it does not vanish, lie in the same plane, and be 
perpendicular to the resultant moment. If, then, the resultant does 
not vanish, a point of the plane can be found for which the result- 
ant moment vanishes. 

99. If all the forces are parallel, the moment of each of them, 
for any point, lies in the plane which is drawn through the point 
perpendicular to the forces. But the resultant of the parallel and 
equal forces, acting at the point, has the same common direction 
with them, and is, therefore, perpendicular to the resultant moment. 
If, then, the resultant does not vanish, a point can be found for 
which the resultant moment vanishes. 

Hence, if all the forces of a system lie in the same plane, or if they are 
all parallel to each other, their tendency to produce translation or rotation is 
equivalent, either to that of a single force, or to that of a couple of forces. 

100. If of any system offerees, and for a point 

Mis the resultant moment, 

R the resultant of equal and parallel forces acting at 0, 
M p and R p the projections of M and R upon the direction 
ofp, 

and if the same letters accented denote the same quantities for the 
point 0', and if - 

x, y, and z are the rectangular coordinates of 0' with reference 

to 0, 

the value of the moment of the forces for either of the axes 
passing through 0' is, 

M , x ^=M x — zR y -\-yR z . 



— 43 — 

But if the direction of the axis of z is assumed to be the same with 
that of B, these moments become 

M' x = M x +yR, 
My = M y — wR, 

The coordinates of the points, for which the resultant moment has 
the same direction with the resultant, are 

M X __ My 

101. The number of forces which is required to produce any 
of the special effects of a given system of forces, is usually much 
less than the whole number of those which actually concur in 
their production. The mode of analysis, by which the requisite 
forces may be ascertained, is, in most cases, quite as simple as that 
by which the effects of rotation and translation have been investi- 
gated. 

III. 

GRAVITATION, AND THE FORCE OF STATICAL ELECTRICITY. 

102. Gravitation is, among all the forces of nature, conspicuous 
for its universality, and the grandeur of the scale upon which it is 
exhibited. 

Each f article of matter is an elementary centre of action for the force 
of gravitation, and all the level surfaces for each particle are spherical 
surfaces, of ivhich the particle is the centre. The value of the potential for 
any particle, is inversely proportional to the distance from the particle, and 
for different particles it is proportional to the mass of the particle. 

103. Another force which seems to be equally universal with 
gravitation, and of which gravitation has been, perhaps justly, 



— 44 — 

regarded as a residual force, and which is subject to the same law, 
in respect to distance from each elementary centre of action, is that 
of statical electricity. This force, however, is endowed with duality, 
and consists of tivo forces, of which one has a positive, and the other a 
negative potential. Both forces are usually combined with equal 
intensity, in the same centre of action, so as to neutralize each 
other's influence, and thus lie dormant. With each of these the poten- 
tial is positive in reference to electricity of the other land, and negative ivith 
reference to that of the same kind. The tendency to motion, arising 
from one kind of electricity, is exactly equal and opposite, then, to 
that which arises from the action of an equal intensity of the other 
kind, distributed in the same way. 

104. The action of electricity upon the mass of a particle 
is indirect ; the direct action is upon the electricity associated 
with the mass. In most bodies the electricity yields with more or 
less facility to this action, leaves the particle with which it is 
originally combined for another particle, and finally assumes such a 

form of distribution within and upon the body, that the tendency to motion 
shall nowhere exceed the resistance to motion. Bodies in which there is 
no resistance to the motion of electricity are called perfect conductors; 
while those in which the resistance is infinite are called, perfect non- 
conductors. 

105. Let 

dm denote the mass of a particle of matter in the case of 
gravitation, or the value of its potential at the unit of 
distance, in the case either of gravitation or elec- 
tricity ; 

da, the element of volume of the mass ; 

/-, the density of the matter, in the case of gravitation, or 
the intensity of the force of electricity, compared 
with the unit of intensity ; 



— 45 — 

/, the distance from the particle ; 
dS2, the value of the potential for the particle ; 

the expression of the potential for the particle is 

, r -> dm kda 

m = T = T- 

The general value of the potential for the whole body is 

°=fJ=SJ 

106. With reference to a system of three rectangular axes, 
let 

x, y, z, be the coordinates of the point in space, for which the 

potential is £2, and 
£;, i], l, those of the particle. 

Adopt also the functional notation 

The derivatives of/ and/ -1 are 

D x f=COBf = — — . 



~" P ~~T' 



D- = — -Df — 



Hence 



x f ~ P xJ ~~ P ' 
Dl± = -±Dlf+?r z {D x fy = ~ sin2 y 2cos " ; 

_ — 1 + 8008^ 

— J3 -■ 



1 — 3 + 3^cos 2 ^ ft 

V J~ " ~P~ ~~ ' 



— 46 — 

and, therefore, 

pd£2 = Q, 

This last equation, which is called Laplace's equation, only 
applies to that extent of space for which the derivatives of the 
potential are continuous functions, that is, where there are no 
centres of action ; but, where there are centres of action, it requires 
a modification which will soon be investigated. The integration of 
this equation, combined with peculiar considerations in special 
cases, gives the value of the potential for all the problems of 
gravitation or statical electricity. 

107. The tendency to motion, resulting from the gravitating or 
electrical action of a particle of matter, being normal to the level surface, is 
directed in the straight line drawn to the particle. Its intensity is the 
derivative of the potential, and expressed by the equation. 

The force of the gravitating or electrical action of a particle of matter, 
is, therefore, inversely proportional to the square of the distance from the 
particle. It is attraction in the case of gravitation, or behveen electricities 
of opposite kinds, and repulsion betiveen electricities of the same land. 

ATTRACTION OF AN INFINITE LAMINA. 

108. The investigation of the potential of a lamina of uniform 
density, and included between two infinitely extended planes, is 
simplified by the consideration, that it must have the same value 
for all points of space which are at the same distance from either 
surface of the lamina. Because all such points are similarly situ- 
ated with reference to the lamina, on account of its infinite extent. 
Hence, if either surface of the lamina is adopted for the plane of yz, 



— 47 — 

the derivatives of the potential, with reference either to y or z, 
must vanish, and Laplace's equation becomes 

The integral of this equation gives the value of the potential, 
for a point external to the lamina, or upon its surface, 

in which A and B are arbitrary constants. 

109. The level surfaces are the planes determined by the 
equation (47 7 ), when £2 is the constant value of the potential for 
the level surface. 

110. The action of the lamina upon any external point, is in 
a direction perpendicular to either surface, and its force of attraction 
or repulsion is constant upon all points, for it is given by the equation 

111. The values of A and B in any special case must be 
ascertained by direct integration. The integration indicated in 
(45 8 ), gives an infinite value of the potential, whereas the integra- 
tion of its derivative, with reference to x, gives A itself, in a finite 
form, which shows that the infinite portion of the potential belongs 
to B. The integration for finding the derivative of the potential is 
effected by putting 

Q=fsm*, 

= the projection of/ upon the plane of yz. 
a ■=■ the thickness of the lamina ; 
whence 

f={x — £)sec*, 

o = (x — £)tan^, 



— 48 — 

da = Qdod y d'E,, 

= (x — I)'- sm x f sen" dUHl 

=-£fzP sia ?- 



= — 2jzaJc = A. 



This value of A corresponds to a positive value of x, but for a nega- 
tive value of x its sign must be reversed. 

112. For a point situated within the lamina, a plane may be 
drawn through it parallel to the superficial planes, and dividing the 
lamina into two partial laminse, of which the thicknesses are x and 
a — x. Hence, the value of the derivative of the potential is 

D X S2 = — 2nJcx -\- 2nk{a — x) 
= 2nk(a — 2x). 

poisson's modification of laplace's equation for an intepvIor point. 

113. The modification which is required of Laplace's equa- 
tion, in order that it may be applicable to any point of an acting 
mass, must be the same for all cases. For it would not be needed, 
if the point of action were contained within any extent, however 
small, of void space. It depends, therefore, exclusively upon the 
infinitesimal portion of matter at the point, and is unaffected by 
any variations in the form and extent of the acting body. It need 
be investigated, then, in only a single case. Now the derivative 
of (48 16 ) gives 

D 2 X £2 = — 4:7i7e, 

which substituted in Laplace's equation gives for an internal point 



— 49 — 

of the infinite lamina, 

p£2 = — inJc 



05 



"which is, therefore, the required modification of this equation. 
This modified equation, in which Ic , denotes the value of 7c at the 
point of action, is applicable, as remarked by Sturm, even when the 
point is exterior to the body. This same geometer has observed 
that, by supposing the value of Jc gradually to shade off from its 
value within the hodiy to zero, this graduation occurring within an 
infinitely small extent, so as not sensibly to interfere with the 
actual phenomena of nature, the potential and its differential coeffi- 
cients may become continuous functions. It must be further 
observed, however, that this imaginary graduation must extend 
throughout all space, although k must have an infinitesimal value 
where there is no portion of active force ; for if it were to vanish 
throughout any finite portion of space, however small, the reason- 
ing of § 69, would prove that all the derivatives of the potential 
were not finite and continuous. 



ATTRACTION OF AN INFINITE CYLINDER. 



114. The investigation of the potential of an infinite cylinder 
is simplified by the consideration that its value must be the same 
for all points situated upon the same straight line parallel to one of 
the sides of the cylinder. If this direction is adopted for the axis 
of s, the derivative of the potential, with reference to z, must 
vanish, and Laplace's equation becomes 

r£2 = {Di+D%)n = o. 

The integral of this equation is 

S2 = $(* + ysTi) + ®i(x — yvQ, 

7 



— 50 — 

in which *$ and 9^ are arbitrary functions, and must be determined 
for each case by special considerations. 

115. The level surfaces are the cylindrical surfaces, of which 
(49 30 ) is the general equation, if £2 has the constant value belonging 
to that surface. 

116. The attraction in the direction of the axis of x is 

in which the accents denote the derivatives of the functions, with 
reference to their explicit variables. 

The attraction in the direction of the axis of^ is 

D y £i = [#'(* +yvQ - ^'0 -ytfTx)]*Ci. 
The whole action is, then, 

s/[(B x y^(n y y]n = 2^[W(x+y^).^(x-y^[)^ 

117. When the point of action is so far from the cylinder that the 
square of the linear dimensions of the base can be neglected, in comparison 
ivith the square of the least distance of the point from the cylinder, the 
problem can be greatly simplified. 

Find in this case a line parallel to the axis of z, of which the 
coordinates a and b, with reference to the axes of x and y, are 
determined by the equations 



f (£ — a) = = f I — am, 

J Or} — b) = 0=l 1] — bm. 
m \J in 



This line may be called the axis of gravity of the cylinder, and 
its position is wholly independent of the directions of the axes of 
x and y. For the conditions by which this axis is determined will 



— 51 — 

give, with regard to any other axis of x, with reference to which 
the notation is distinguished by the subjacent numbers, 

If the axis of gravity is, then, assumed for the axis of z, the 
equations (50 25 _ 2 o) become 

or 

118. Since, from the nature of the cylinder, the functions 
which are here to be integrated are independent of C, these 
equations give 

119. Let the perpendicular from the point of action upon the 
axis of z be assumed for the axis of x, and let 

/ be the distance of the point of action from the projection of 

any particle of the cylinder upon the axis of z, 
o the distance of the particle from the axis of z. 

The conditions of the problem under consideration give 

- :== — (l 4- — ^ = — -t- ^1 ' 

J / o \ Jo/ Jo Jo 

J mj J mJo JmJ 

Jmfo J tfoJ %J ri Jmfo' 



— 52 — 

so that the potential is the same as if all the particles of the cylinder ivere 
united in their projections upon the axis of gravity, when the point is at a 
sufficiently great distance from the cylinder. 
120. By letting 

K denote the intensity of the action concentrated upon 
each point of the axis of gravity when the cylinder 
is projected upon it; 

the value of the whole action of this axis is 

oo oo 

•A*^ = — / L Jr= — Kx I , .„ , >.„. g 

CO —CO 

in 

KP x 2K 
= I* cos* = , 

*J/o /o x 



or the potential is 

£2= — 2Klogx-\-B, 

in which the arbitrary constant B is infinite. 

121. When the base of the cylinder is the space which is contained 
hctiveen two concentric circles, the axis of gravity coincides with the 
geometrical axis, the potential is, from the symmetry of the figure, 
the same in all directions from the axis, and its value only depends 
upon the distance from the axis. Let the axes be the same as in 
§§117 and 119, except that the point of action is in the plane of 
x y, but not in the axis of x, and let 

r = the radius vector of the point of action, and 
c = the base of the Naperian system of logarithms. 

The potential is a function of r, and does not involve the inclination 
of r to the axis of x. Hence 

Z>*S2 = 0. 



— 53 
But by (49 30 ) 



n = ®\rc' S 7 + ^C'c " /'> 



whence 



and 



D*& = lW\rc he —W\re he JvCi=0; 
W\rc Jrc = < $[\rc /re 



But the two members of this equation are functions of two different 
and independent variables, which are 

r V — 1 — r V — 1 . 

re and re > 

and, therefore, neither can be contained in the value of the other, 
so that each of them disappears from their common value, which is, 
therefore, constant. With regard to any variable whatever, there- 
fore, this equation gives 



rWr = rW 1 r = A, 
and, by integration, 



&r = &ir=*} 



< $r = A\ogr-\-B, 
< & 1 r = Alogr-^-B^ 

The value of the potential is, then, if the two constants are com- 
bined in one, 

(— \l \ / — X \I \ 

, rc r ~J-\-A\og\rc ' Z+^2 

= 2Alogr + B 2 , 

and the action upon the point is in the direction of r, and its 



— 54 — 
value is 

r 

122. When the point of action is upon the axis, it is plain, 
from the symmetrical nature of the cylinder, that the action is 
cancelled in each direction, and in this case 

whence 

4 = 0. 

For every point within the inner cylindrical boundary of this cylindrical 
shell, the action, therefore, vanishes, and the potential is constant. 

123. When the point of action is without the cylinder, the 
constants are found by the condition that when the distance is very 
great, the value must be the same as that of (52 13 ). Hence 

A = — K, 

that is, the action upon every point, ivithont the circular cylinder, is the 
same as if the ivhole mass of the cylinder were concentrated upon its axis. 

124. No other case of the infinite cylinder is of sufficient 
interest to divert the current of the work from the finite masses 
of nature. 

RELATION OF THE POTENTIAL TO ITS PARAMETER. 

125. The varying value of the potential from one level 
surface to another, depends upon the law of the change of surface, 
and may be represented as a function of a variable, which may be 
called its parameter. Let 

X denote the parameter of the potential, and adopt the func- 
tional notation 

□ = S x {D x f = {D x f + (D y y + (A) 2 - 



55 



The derivative of the potential gives 
DJ2=D 7l P-D x l, 

which is a transformation given by Lame. 

126. For a point of void space, this equation gives 

^ = DMogDM) = — ^ ; 
by which the potential may be determined for given forms of I. 

ATTRACTION OP A FINITE POINT UPON A DISTANT MASS. CENTRE OF GRAVITY. 

127. In every finite mass there is a point called the centre 
of gravity, of which the coordinates are determined by equations, for 
each axis, which are similar to (50 25 _ 2 6)- This point is independent 
of the positions of the axes, for these equations give for any other 
axis 

If the centre of gravity is adopted for the origin of coordinates, 
these equations are reduced to (51 8 _ 10 ). 

128. When the point of action is so far from the attracting 
mass, that the squares of the linear dimensions of the mass may be 
neglected in comparison with the square of the distance of the 
point from the mass, the formula becomes 

f = 2.(* — ty = 2 x {z* — 2*£+|") 



— 56 — 
- f - 4- 2 (- f t) 



m 



that is, the potential of a finite point, for a mass which is so remote that 
the square of the linear dimensions of the body may he neglected, in compari- 
son with the square of the distance of the point from the body, is the same as 
if the body were concentrated at its centre of gravity. 



ATTRACTION OP A SPHERICAL SHELL. 

129. In the case of a shell of homogeneous matter, contained 
between the surfaces of two concentric spheres, the value of the 
potential must, from the symmetry of the figure, depend exclu- 
sively upon the distance from the centre ; and for the same reason 
this centre is the centre of gravity. If the centre is adopted for the 
origin of coordinates, the parameter may be assumed to be the 
radius vector, or any function of it. Putting, then, 

derivation gives 

D x l = 2x, 

PX = Q. 

Hence, (55 8 ) becomes 

The integral of this equation is, by the introduction of the arbitrary 
constants A and B, 

S2 = B-i = B-^. 

\ K r 



— 57 — 

130. When the point of action is at the origin, the value of 
the potential is easily obtained by direct integration. Let in this 
case 

() and o x be the internal and external radii of the spherical 
shell, 

?n and m x the masses of two homogeneous spheres of the same 
density with the shell, and of which the radii are respect- 
ively ^) and q 1 ; and 

dip the elementary solid angle of which the vertex is at the 
point of action. 

The mass of the shell is 

m = m-L — m = 4 n # (o f — qI), 

and the element of mass 

dm== 7eQ 2 dydQ. 

The value of the potential is, therefore, 

= Hf( Q \ — 9 l) = 2 7tk(Ql — 9 ») 

t/Tp 

__ i /% ™o\ 

131. When the point of action is in the interior void space of the 
shell, the constants of (56 30 ) must have the same values as at the 
origin, where r vanishes. Hence, for this space, the constants are 

,1 = 0, 

J = 2**( 9 J- ? 5) = i(=i-=2J. 

The value of the potential in the interior void space is, therefore, 
constant, and there is no tendency to motion in any direction. 



B = 


= 0, 


A = 


= — m ; 


£2 


m 



— 58 — 

132. For an exterior point, the potential vanishes when r is 
infinite, while for a point at a great distance from the origin, its 
value is, by § 128, the same as if the whole mass were concentrated 
at the origin. The value of the constants in this case are then 



and the potential is 



Any exterior point is, then, attracted by a homogeneous spherical shell, 
precisely as if the ivhole mass of the shell tvere concentrated upon its centre 
of gravity. 

ACTION AND REACTION OF A SURFACE OR INFINITELY THIN SHELL 
OF FINITE EXTENT. CHASLESIAN SHELL. 

133. An infinitely thin shell may be reduced to either of its 
surfaces, upon which all its acting force may be concentrated, and 
the intensity of the action at each point of the surface will be the 
product of the corresponding intensity of the force of the shell, 
multiplied by the thickness of the shell, and the element of the 
surface must be substituted for the element of volume of the shell. 
Let then, 

do be the element of the surface, 

N the exterior direction of the normal to the surface, 

Jc the concentrated intensity of action at any point of the 

surface, 
dip the elementary solid angle subtended by the element of the 

surface at the point of action ; 

the expression of the element of the surface is 



do =f 2 dy sec j£. 



— 59 — 
Hence 

7 7 C0S f 1 7 

— /cctip = j^-lcdc) . 

The second member of this equation denotes the action 
exerted by each element of the surface in a direction normal 
to the surface, and towards the interior of the surface. If, there- 
fore, the intensity of action is constant over the surface, the action 
normal to the surface is proportional for each element of the 
surface, to the solid angle subtended by the element, and the total 
amount of the action, normal to the surface, exerted by any continuous extent 
of the surface, is proportional to the whole solid angle subtended by the 
boundary of the surface. 

134. If the surface is a plane, the direction of the normal is 
invariable, and the total amount of normal action exerted by any 
portion of the plane is the same with the projection of the ivhole action 
of this portion of the plane upon the perpendicular to the plane, which is 
therefore proportional to the solid angle subtended by the portion of the 
plane at the point of action. 

135. If the surface returns into itself so as to include a space, ivhich 
is called a closed surface, and if the point of action is situated within the 
inclosed space, the ivhole angle subtended is the entire extent of four right 
angles ; whereas, if the point of action is exterior to the closed surface, the 
whole angle vanishes ; but it is tivo right angles when the point is upon the 
surface. For, however the point of action is situated, if a line is 
drawn from it so as to cut the surface more than once, the 
successive angles which the line makes with the exterior normal, 
will be alternately obtuse and acute as the line cuts into the 
surface or out from it. The last angle, or that of which the vertex 
is most remote from the point of action will always be acute. The 
normal actions of two successive elements, therefore, upon the same 
line, and which subtend the same solid angle, are equal, but of 



— 60 — 

opposite signs, so that they cancel each other's effect in the total 
sum of the normal forces. But if the point of action is without the 
surface, the first angle is obtuse upon each line, and as the last 
angle is acute, the whole number of intersections is even, and each 
normal elementary action is cancelled by another, and the whole 
sum vanishes. If the point of action is within the surface, the first 
angle is acute, if there is more than one ; and there are an odd 
number of intersections for every direction in which a line can 
be drawn ; for each direction, therefore, one, and only one, normal 
action remains uncancelled, which is proportional to the elemen- 
tary solid angle ; and the whole sum is that of the entire extent 
of four right angles. But, if the point of action is upon the 
surface, and a tangent plane to the surface is drawn through it ; 
every line which is drawn from the point upon the exterior side of 
the plane must cut the surface an even number of times, if it cuts 
at all, precisely as if it were drawn from an exterior point; but 
every line which is drawn upon the interior side of the plane cuts 
the surface, as if it were drawn from an interior point ; the total 
sum, then, of the uncancelled elementary solid angles includes those 
for all directions which are upon the inner side of the plane, that is, 
it is equal to two right angles. This elegant theorem, given by 
Gauss, is expressed analytically in the form 



I 



4 n for a point interior to a closed surface, 
„-= < 2?r for a point upon the surface, 
for an exterior point. 



136. The expression (59 2 ) represents the component in the 
direction of the external normal to a surface, of the action upon the 
element of the surface of a mass Jc concentrated at the point which, 
in that expression, was the point of action. The integral of this 
expression is the whole amount of such resolved action, and by 



— Gl — 
(60 24 ) its value is 

C — 4:7tk when the mass h is interior to the surface, 
— / — ^h = — / Jc = < — 2nk when the mass k is upon the surface, 

(^ when the mass k is exterior to the surface. 

Neither of these values depends upon the position of the acting 
mass further than it is interior or exterior to the surface or upon 
the surface. If, then, 

Mi = all the mass interior to the surface, 

M u = all the mass upon the surface, 

M e = all the mass exterior to the surface ; 

the expression for the total action of the sum of all the masses upon a closed 
surface, resolved for each element in the direction of the external normal, is 



4:7cM — 2nM. 



u ) 



and if all the masses are exterior to the surface, this sum vanishes. If the 
closed surface is one of the level surfaces of the system of bodies, this sum 
expresses the total attraction of the masses upon the surface. This impor- 
tant theorem is due to Gauss, and, independently to Chasles, in 
almost its full extent, as well as most of the following deductions. 
It is applicable, even if the surface have sharp angles, because the 
extent of surface occupied by such angles is zero. 

137. If the closed surface is one of the level surfaces of a 
system of bodies, but not the outer boundary of a space in which 
the potential is constant, the potential must at each point, by § 67, 
increase in passing from the interior to the exterior or the reverse, 
so that in this case the sum (61 16 ) does not vanish. But the term 
of this sum, which depends upon the mass at the surface, may be 
neglected at will ; for the whole mass of a true geometrical surface 
is absolutely nothing. Hence, every level surface must inclose masses of 



— 62 — 

matter, unless it be the outer material boundary of a space in which the 
potential is constant. 

138. When any masses lie upon the closed surface, the geo- 
metrical surface may, as Gauss observed, be arbitrarily assumed as 
being just exterior or interior to the masses, or passing through 
them. If, therefore, all the masses are so distributed upon a surface that it 
becomes itself a level surface, the potential is constant for all the inclosed 
space, and there is no tendency to motion throughout this space. 

139. Around every point of maximum or minimum potential 
a level surface of infinitesimal dimensions may obviously be drawn ; 
and, therefore, every point of maximum or minimum potential must be itself 
a centre of action, and cannot be a void space. 

In an inclosed space, therefore, no point can be found for which the 
value of the potential exceeds the limits of value which are found upon the 
inclosing material surface; and in no point of unbounded space has the 
potential so great a value as its greatest value upon the exterior surface of 
the finite masses. This inference was drawn by Gauss. 

140. In a system of bodies, of which gravitation is the only force, 
there can be no point of absolute minimum potential. For if about a point 
of maximum or minimum potential, as a centre, an infinitesimal 
sphere is described, there can be no point within the sphere, either 
of maximum or minimum potential, with reference to the matter 
external to the sphere. But, with reference to the matter of the 
sphere itself, the centre must be a point of maximum potential, and, 
therefore, cannot be a point of minimum potential, with reference 
to the combined action of all the masses. 

This theorem is equally applicable to an aggregation of elec- 
tricity, all of which is of the same kind, that is, which is homogeneous 
when the point of action is assumed to be of the opposite kind of 
electricity. 

141. If any extent of level surface is assumed at will as a 



— 63 — 

base, and if trajectories, like those of § 68, are drawn through each 
point of its perimeter, their union forms a canal. The same canal 
cuts a base, like the assumed base, from each level surface which it 
intersects. Of any canal, then, ivhich is not extended so far as to include 
portions of the attracting masses, the attractions upon all the bases are equal. 
For the whole amount of action, resolved in the direction of the 
external normal, at each point of action upon the closed surface, 
formed by the faces of the canal and the two terminating bases, 
vanishes, because there is no included mass. But there is no action 
perpendicular to the faces, that is, in the direction of the level sur- 
faces ; whereas the whole action upon the bases is normal to them. 
The actions upon one base are in the directions of its external 
normals, while those upon the other base are in the directions of 
the internal normals ; but these actions balance each other in the 
algebraic sum, and, therefore, their absolute values must be the 
same. This theorem belongs to Chasles, but the brief demonstra- 
tion is original. 

142. In the following simple view of this whole subject, many 
of its propositions are condensed into a small compass. Each centre 
of action may be regarded as a fountain from which a stream is 
perpetually flowing in every direction, with an amount of discharge 
proportioned to the intensity of the action. The quantity which 
flows from each centre, for an instant, through any given elemen- 
tary surface, may easily be shown to be in exact proportion to the 
force with which the surface is attracted by this centre perpendicu- 
larly to itself and against the current ; and that which is true for 
each centre is also applicable to the combined action of all the 
centres. Upon a space, then, in which there is no spring, the 
amount which is flowing out must constantly be equal to that which 
is flowing in ; while from a space which contains springs, the amount 
which is discharged must exceed the inward flow by all which is 



— 64 — 

supplied by the fountains. These propositions are equivalent to 
those of § 136, and it may be shown by an easy argument that 
Laplace's equation, with its modification, is merely the same propo- 
sition applied to the element of space. 

By the additional hypothesis, that, to preserve the uniform 
flow of the stream, its density must decrease in each element of 
the stream with the distance from the origin, so as always to be 
inversely proportional to the distance from the centre, the potential 
represents the density of the combined streams, and the level 
surfaces become surfaces of equal density. The aggregate current 
of the combined streams is also equivalent to a single current in a 
direction perpendicular to the level surfaces, and having a velocity 
proportionate to the rate of decrease of density. But this is the 
well known law of the propagation of heat, when there is no 
radiation, and hence arise the analogies between the level and 
isothermal surfaces, and the identity of the mathematical investi- 
gations of the attractions of bodies and of the propagation of heat 
which have been developed by Chasles. 

143. If an infinitely thin homogeneous shell is formed upon each level 
surface of a system of bodies, having at each point a thickness proportional 
to the attraction at that point, the portion of either of these shells, which is 
included in a canal formed by trajectories, bears the same ratio to the ivhole 
shell, ivhich the portion of another shell included in the same canal bears to 
that shell, provided there is no mass included between the shells. For if the 
bases of the canal are infinitely small, they must be reciprocally 
proportional to the intensities of the actions upon them, because the 
whole amount of action upon the different bases is the same. But 
the thicknesses of the shells are proportional to the intensities of 
action, and, therefore, the products of the bases multiplied by the 
thicknesses, or the volumes of the portions of shell included in the 
same canal, bear a constant ratio to each other. Since the ratios 



— 65 — 

are constant the infinitesimal volumes may be added together, and 
their sums, which are the volumes included in a finite canal, are in 
the same ratio, and these sums may even be extended so as to 
include the whole of each shell. Hence the volume of each portion 
is the same fractional part of the volume of the shell to which it 
belongs ; and, as each shell is homogeneous, the mass of each por- 
tion is the same fractional part of the mass of the whole shell. The 
conception of these shells, and the investigation of their acting and 
reacting properties was original with Chasles, and it will be con- 
venient, as it is appropriate, to designate them as Chaslesian shells. 

144. The volume or mass of a Chaslesian shell has a simple 
ratio to the attracting mass included within it, dependent upon its 
own density and thickness. For each infinitesimal element of its 
volume or mass is proportional to the product of the element of the 
surface multiplied by the thickness of the shell, and the thickness at 
each point is proportional to the attraction at that point. The sum 
of all the elements, therefore, of either volume or mass, that is, the 
whole volume or mass, is proportional to the sum of all the attrac- 
tions upon the whole surface. But, by § 136, the sum of all the 
attractions upon the surface is proportional to the included mass, if 
there is no mass at the surface. If, then, 

p is the volume of the shell, 
k its density, 

h the modulus of its thickness, or the thickness which corre- 
sponds to the unit of attraction ; 

this ratio is included in the equation 



H Ten 



KM khM 



4:71, 



145. If a Chaslesian shell which is ivholly external to the acting 
masses of the si/stem is assumed to be itself the attracting mass ; 

9 



— 66 — 

1. The potential of the shell is constant for all interior points, there 
is no tendency to motion within it, and its own outer surface is its level 
surface ; 

2. Its external level surfaces are the same as those of the original 
masses of the system, and the attraction of the shell upon a point external to 
itself has the same direction as the attraction of the original masses. 

To demonstrate these propositions, let 

S2 S be the potential of the shell for any point, and 
£2 the potential of the original masses for each point of the 
shell ; 

the value of the element of the potential of the shell is 



Hence, 



In passing along the canal of the trajectories to another shell, 
the ratio of d/.i to p is, by § 143, constant, whence 

j-. dSi s _ hd^D^f 

But 

DJ = D x ND N f = - B.JYcosf, 
df.i = lido D N S2 ; 
and, therefore, 

d\i D x f = — h do D N S2 D x JVcos * = — hcla D^ £2 cos y , 

V ^ — - —^ TV 
The integral of this equation for the whole surface of the 



d£2 s 


Jcdfi 


dSi, 


Jcdfi 


[i 


Vf 



— 67 
shell is 



n Si, l-hD x Sl C c° s / 



1 (i 



J a J 



1. For an internal point this equation becomes, by §§ 135 
and 144, 

n Si, 4akhD ? Si kD^Sl 

the integral of which is 

Q _ kflSi 



M 



5 



to which no constant need be added, because, when the dimensions 
of the shell are infinite, £2 and S2 S both vanish, since all the points 
of action are infinitely remote from the centres of action. This 
equation expresses that the potential of each shell has the same 
value for all internal points, and, therefore, there is no tendency to 
motion within the shell, and the surface of the shell must be level, 
with reference to its own action. 

2. For an external point, the equation (67 2 ) becomes, by 
§135, 

Hence, by integration, 

— = a constant, 

which constant, however, depends for its value upon the position of 
the points of action ; but since it has the same value for all the 
shells to which the point is external, the potential is constant for 
the same series of points external to one shell for which it is 
constant through the action of another shell ; that is, all the shells 
have the same external level surfaces. But the external level 
surface, which is nearest to any shell, differs infinitely little from 



— 68 — 

the level surface of the shell itself, and, therefore, the surface of 
each shell is a level surface for every included shell. Hence, the 
external level surfaces of a shell are the same with those of the 
original masses, and the attraction of a shell upon an external point 
has the same direction with the attraction of the original masses, 
and is normal to the level surface passing through the point. This 
theorem is due to Chasles. 

146. Every infinitely thin shell, of which the surface is level, from the 
action of the shell itself, must be a Chaslesian shell. For, if another shell 
is constructed upon this level surface, which is the negative of the 
Chaslesian, one, namely, which is repulsive, instead of being attrac- 
tive, or the reverse, and the whole mass of which is equal to that of 
the given shell, the two shells, having the same level surfaces, 
exactly cancel each other's action throughout all space. The 
elements of mass of the two shells must then be absolutely equal, 
but of opposite signs at every point. For, if they were unequal at 
any point, that point might be made the centre of an infinitely thin 
circular element of the combined shells. From the symmetry of its 
figure, a level surface for the action of this element alone might be 
made to pass through its perimeter, and which could inclose no 
other mass than the element itself. But such surface cannot be 
level for the remainder of the combined mass of the two shells, and, 
therefore, the value of the potential upon this surface for the 
combined masses of both shells, including the circular element, 
cannot be constant. This want of constancy in the potential is 
contradicted by the fact that the shells balance each other's action 
everywhere. There cannot, therefore, be any such want of con- 
stancy, nor any point for which the element of mass of the given 
shell is not absolutely equal to that of the Chaslesian shell, although 
it is of a contrary sign. But reversal of the sign of the action of 
the mass does not interfere with the Chaslesian characteristic of the 
shell. 



— 69 — 

147. Two Chaslesian shells, which are constructed upon the same 
surface, only differ in their density and their modulus of thickness. For 
the density of either of them may be increased or decreased until 
the value of its potential at the common surface shall be equal to 
that of the other shell. If, then, its action be reversed, the value 
of the potential for the combined shells will be zero both at the 
surface and at an infinite distance from the surface ; and it cannot 
have any other value in the intermediate space, otherwise, there 
would be points or surfaces of maximum potential exterior to the 
acting masses. The combined surfaces have, therefore, neither 
external nor internal action, and the reasoning of the preceding 
article demonstrates that the component shells are identical, except 
in regard to their signs. 



ATTRACTION OF AN ELLIPSOID. 



148. An infinitely thin homogeneous shell, of ivhich the inner and 
outer surfaces are those of similar, and similarly placed, concentric ellipsoids, 
is a Chaslesian shell. For, if upon the longest axes of these ellipsoids, 
as diameters, two concentric spheres are constructed, each sphere 
may be compressed into the corresponding ellipsoid, by reducing 
all the coordinates from the centre, as origin, parallel to either of 
the two shorter axes of the ellipsoid in the ratio of the longest axis 
to this shorter axis. But all points, which are originally in the 
same straight line remain upon a common straight line after this 
uniform compression ; and all distances which are measured in the 
same direction are reduced in a common ratio. But the thick- 
nesses of the spherical shell, measured upon any straight line at the 
two points where this line cuts the shell are equal ; so that the 
thicknesses of the ellipsoidal shell, measured at the two points 
where the reduced line cuts this shell, are also equal. If, then, at a 



— 70 — 

point assumed at will, as the vertex, within the ellipsoidal shell, an 
infinitesimal cone is constructed and extended in each direction 
from the vertex, till it intersects the shell, the relative masses of the 
two included portions of the shell are proportional to the squares of 
their distances from the vertex; and, therefore, their attractions 
upon the vertex are equal, but in opposite directions. Hence, the 
action of any portion of the shell upon an internal point is balanced 
by the action of the opposite portion, and there is, consequently, no 
tendency to motion within the shell from its own action. The 
surface of the shell is thus proved to be a level surface, in respect to 
its own action, and, by § 146, it can be no other than a Chaslesian 
shell. 

149. This proposition may be enlarged to a theorem given by 
Newton, for a finite shell, of which the inner and outer surfaces are 
those of similar and similarly placed concentric ellipsoids. Such a 
shell may be called a Newtonian shell, so that the infinitely thin 
Newtonian shell is a Chaslesian ellipsoidal shell. But the New r - 
tonian shell may be subdivided by similar and similarly placed 
concentric ellipsoidal surfaces into an infinite number of Chaslesian 
ellipsoidal shells, each of which is inactive with reference to an 
internal point. Hence, the whole Newtonian shell exerts no action upon 
an internal point. 

150. An ellipsoid may be converted into any other similar, 
and similarly placed, concentric ellipsoid by a process similar to that 
by which the sphere in § 148 was changed to an ellipsoid ; that is, 
by increasing or decreasing the coordinates of each point, taken 
from the centre as origin, and parallel to either axis, in the ratio of 
the corresponding axes of the two ellipsoids. The points of the two 
ellipsoids, which correspond in this process, have been called by 
Ivory corresponding points. By this process, any Newtonian shell 
may be converted into another concentric and similarly placed 



— 71 — 

Newtonian shell, and at the corresponding points there will be 
corresponding elements of volume. 

151. The corresponding elements of volume or mass of two corre- 
sponding Neiotonian shells are proportional to the volumes- or masses of the 
shells. For if 

A x , A y , A z are the semiaxes of the outer ellipsoidal surface of 

one shell, 
B x , By, B z those of its inner ellipsoidal surface, 
a its volume, 
m its mass, and 
n the ratio of either axis of the inner surface, divided by the 

corresponding axis of the outer surface ; 

and if the same letters accented denote the same quantities for the 
corresponding shell, the construction of the shells gives for each 
axis 

B x = nA x , 





X X 

A x A x 


and 


11 = ri ; 


and by differentiation, 






dx A x 
dx 1 A 7 ' 



The volumes and masses are by well-known theorems of 
geometry 

m = Jeo = jn k (A x Ay A z — B x B y B z ) 

= i7ik(l — n 3 )A x A y A z , 
m' = 7<f o' = j 7i k' (I — n s ) A x A y A z . 

The ratios of the elements of volume and mass are, then, 

da dxdydz _A x A y A z a 



da' dx'di/dz? A' x A' y A 



' A' <t" 

X ^M-yJ3. z U 



— 72 — 

dm kda ha m 
dm! hda' ho' m' ' 

152. If the older surfaces of two corresponding Newtonian shells have 
the same foci, their inner surfaces must also have the same foci. For if 

e 2 is the difference of the squares of the corresponding axes of 
the outer surfaces, 

the condition of the identity of foci gives the equations 

p 2 J 2 /j'2 /12 A'Z /12 A'l 

c J± x J± x J± y Ji y xi z Ji z . 

Hence, for each axis, there is the equation 



so that the foci of the inner surfaces are also identical. 

153. If the radius vector, from the centre of any point of an ellipsoid, 
is projected upon the radius vector of another ellipsoid ivhich has the same 
foci, and if the radius vector of the corresponding point of the second ellipsoid 
is projected upon that radius vector of the first ellipsoid, ivhich corresponds in 
direction to the projection in the second ellipsoid, the tivo projections are 
inversely proportional to the radii vectores upon ivhich they are projected. 
For if 

o is the radius vector of the first ellipsoid upon which the 

projection is made, and 
£, rj, t, are the coordinates of the extremity of q ; 

the equations of the corresponding points give, for each axis, 



whence 





_ r x _ x' t 

■A x M-x A x 




I— I 

x x n 



or 



►7^ 

to 



lx'=l'x. 



But if 



p is the projection of/ upon q, and 
p the projection of r upon (/, 



these projections are 



whence 



, r' ^ a/$ 2: r (^£) 
» = rcos = 2l x — = — - — -, 

1 Q Q Q 



P' Q 



154. The difference of the squares of the radii vector es from the 
centre, of two corresponding points upon the surface of two ellipsoids which 
have the same foci, is equal to the difference of the squares of their semiaxes. 
For the equations of these surfaces are 



~2 ~/2 

2 — = 1, 2 —= 1 

•At: SI* 



The difference of the squares of two corresponding radii 
vectores for points at the surface, is 



r' 2 



= 2 M (** — af*j=Z t [z*(l — ^)] 






155. The distance of any point upon the surface of an ellipsoid, 
from a point upon the surface of another ellipsoid which has the same foci, 
is equal to the distance of the two corresponding points of the ellipsoids from 
each other. For if 

10 



— 74 — 

/ is the distance of the point of which 
x, y, z, are the coordinates, from the point of which 
£', if, 'Cf, are the coordinates, and 
f the distance of the corresponding points ; 

the values of these distances become, by (73 9 _ 10 ) and (73 24 ), 
/2 = r 2_j_ (/2 _2 9 y 
/* = /> + Q *-2 QP 



r 



,2 



+ o' 2 + e 2 -2o>' 



2 I , '2 O , ' ' ^2 

whence 

156. The external level stir faces of an ellipsoidal Chaslesian shell are 
those of ellipsoids which have the same foci ivith the order surface of the 
Chaslesian shell. For if 

£2 C is the potential of the given shell for any point of the 
external ellipsoidal surface of the same foci, and 

£l f c the constant value of the potential of the corresponding 
Chaslesian shell, constructed upon the external ellipsoidal 
surface, for any internal point, and, therefore, for any 
point of the surface of the given shell ; 

the equations (72 x ) and (74 12 ) give 
O 



J m J J in J 

" c Jm'f J m mf mjmf m c 



The value of S2 C is, therefore, constant for all points of the 
surface of the external ellipsoid, so that this is one of the level 
surfaces of the given shell. 

157. The attractions of two corresponding Newtonian shells, ivhich have 



— 75 — 

the same foci, upon an external point, have the same direction, and are propor- 
tional to the masses of the shells. For the infinitely thin shell, this 
proposition is a simple corollary from (74 26 ). But the finite shells 
can be subdivided into corresponding infinitesimal shells, and the 
masses of the corresponding elementary shells will be proportional 
to the masses of their respective finite shells. The attractions of 
the corresponding elementary shells upon an external point, there- 
fore, coincide in direction, and are proportional to the masses of the 
shells; and, therefore, the components of all the corresponding 
attractions have the same common ratio, and coincide in direction. 
But the components of all the elementary attractions constitute the 
attractions of the finite shells themselves. Several special cases of 
this theorem were first given by Maclaurin, but the general form 
was first demonstrated by Laplace, and afterwards more rigorously 
by Legendre, and it includes the case in which the inner surfaces are reduced 
to the central point, and the shells become ellipsoids, having the same foci. 

158. The attraction of any Chaslesian shell upon a point at its 
surface is, from its construction, perpendicular to the surface, and 
proportional to the thickness of the shell at that point. The attrac- 
tion upon the whole surface is, therefore, proportional to the mass 
of the surface, which corresponds to § 136. Hence, if 

dN is the thickness at any point, and 

p the perpendicular from the centre upon the tangent plane at 
that point, 

the attraction of the ellipsoidal Chaslesian shell at the point is 

knkdN= in]idro,o$ N 

r 
a 7 dr r 

= 4:7T/c — rcos 

r p 

= 4:7tkp~. 



— 76 — 

The component of this action in the direction of the axis of x is 

, 7 dA x jsr 
4 7T kp -j- cos . 

If, moreover, the equation of the ellipsoid is 

Z = ^.£ — 1 = 0, 

the general theory of contact gives 

2 X {xD x L) 

N D r L pD T L 

cos 



Hence, 



x "s/O^) 2*(xD x L) 
2 x xD x L = 2Z x ^ 2 =Z, 



ai; 



N px 



and ^Ae attraction in the direction of the axis of x of the ellipsoidal Chasle- 
sian shell upon a point at its surface is 

A.nJcp 2 x-~. 

159. The attraction of an ellipsoidal Chaslesian shell upon 
any external point is obtained by describing the corresponding 
Chaslesian shell, for which this point is upon the outer surface, and 
the attractions of the two shells for this point have the same direc- 
tion, and are proportional to their masses ; so that the attractions 
in any direction are proportional to the masses. If, then, the 
accented letters refer to the outer shell, the attraction of the inner 



— 77 — 

shell is 

, , a , dA' r . 7 a p' 2 dA x 

4:71 IC—p A X—rpr= 4 71 kX-, .,„ . . 

a 1 A 'J o A x A x 

160. The condition that the outer surface of the exterior shell 
passes through the attracted point, is expressed by the equation 

This is an equation of the third degree when it is reduced to 
its simplest form. But there are two other surfaces which can be 
drawn through the given point, and which depend for their defini- 
tion upon the solution of the same equation. They are two 
hyperboloids, both of which have the same foci with the outer 
surface of the inner shell, one of which is a bipartite, and the other 
an imparted hyperboloid. For each of the hyperboloids e 2 is 
negative, and its absolute value, independent of its sign, is contained, 
in the case of the imparted hyperboloid, between the squares of the 
mean and least axes of the given ellipsoid, and, in the case of the 
biparted hyperboloid, between the squares of the mean and greatest 
axes. 

161. The points in which all the ellipsoids, which have the same foci, 
are cut by the common intersection of the tivo hyperboloids which have the 
same foci, are corresponding points. For if 

t 2 is the value of — e 2 for either hyperboloid, 

the equation of the hyperboloid for the points of intersection with 
the ellipsoid is 

X 2 

2 — - — = 1 

If the equation (77 7 ) of the ellipsoid is subtracted from this 
equation, the remainder divided by e 2 -j- t 2 is 



— 78 — 



1 ( e 2_^ £ /2 )a; , 



V v fc ~T b >±. y Z 

■£-< ,, y I J I ON / J 1 / 0\ -"^ <r / A •>. I OX /• A ■> I ->\ \J • 



in which #', ,/, and / are accented, in order not to interfere with 
the notation which has been adopted for the corresponding points, 
and which gives for each axis 



A x ~ A' x ~ i/QAl + ay 
The substitution of these equations in (78 a ) reduces it to 



v 



7K = 0; 



*A*(A>— *'*) 
the product of which by t' 2 , added to (76 6 ), is 



= 1. 



~*A%{A% — J*) x Al — s'*— ^> 

which expresses that the point (x,y,z) is upon the surface of the 
hyperboloid, and, therefore, all the corresponding points are upon 
the surfaces of both hyperboloids. 

162. The hyperboloids and ellipsoids which have the same foci, inter- 
sect each other perpendicularly. The conditions that two surfaces of 
which the equations are 

Z=0, and2/=0, 

intersect each other perpendicularly is expressed algebraically by 
the equation for each point of the line of intersection, 

2 X (D X ZI) X L')=0. 

But for the hyperboloids of equation (77 28 ) and the ellipsoid of 
equation (76 6 ) this condition becomes 

*Ai(Ai-s'*) u > 
which is the same with the equation already given in (78 10 ). This 



— 79 — 

same demonstration may be applied to the condition of the perpen- 
dicularity of the hyperboloids, if A\ is diminished by t' 2 , and a' 2 is 
changed into the difference of the squares of the semiaxes of the 
two lrvperboloids. 

163. It follows from these two theorems, which are derived 
from Chasles, that each normal transversal to the ellipsoidal surfaces of 
level is the line of intersection of two hyperboloids ivhich have the same foci. 

164. The lines of intersection of these three surfaces are, upon 
each surface, the lines of greatest and least curvature, for they are a 
special case of the theorem demonstrated geometrically by Dupin, 
that the intersections of three surfaces ivhich cut each other at right angles 
at and infinitely near their common point of intersection, are their lines of 
greatest and least curvature at this point. To demonstrate this theorem, 
let the three normals to the three surfaces at the common point of 
intersection be assumed for the axes of rectangular coordinates, and 
let 

be the equation of the surface, which is perpendicular to the axis of 
x. This condition gives for either of the other two axes 

D,£ x =0, 

in which equation x, y, and z may be mutually interchanged, except 
that the same axial letter must not be repeated in the equation. 
Those equations satisfy of themselves the condition (78 25 ) of per- 
pendicularity of these surfaces at the point of intersection. But the 
intersection of any two of these surfaces coincides with the axis 
wdiich is the intersection of their tangent planes for an infinitesimal 
distance, and the two surfaces are perpendicular to each other for 
this distance. Hence, each pair of surfaces gives an equation of the 
form 

D z {D x L x D x L y + D v L x D y L y + D z L x D z L y ) = 0, 



— 80 — 

which is reduced by (79 20 ) to 

D X L X D X)Z L y -\- D y L y D yz L x = 0. 

The other surfaces give the corresponding equations 

D v L y D 9t x L z -\- D Z L Z D z x L y == 0, 
D Z L Z D Z >y L x -\- D x L x D xy L z = 0. 

The sum of the products obtained by multiplying the first of 
these equations by D Z L Z , the second by — D X L X , and the third by 
D y L y is 

2D y L y D z L z Dl z L x = 0, 

and the corresponding similar equations are obtained by advancing 
each letter to the following letter of the series, x, y, z, and x. But 
the factors D X L X , D y L y , and D Z L Z , are not zero, and, therefore, 
these equations may be reduced to 

L ) y }Z L x = D xz L y = D x y L z = 0, 

which are the well-known conditions that the directions of the axes 
of x, y, and z respectively coincide with those of the lines of greatest 
and least curvature of the three surfaces at the origin. 

165. The remarkable relations of these surfaces might be still 
further extended, and if it were worth while to investigate the 
attractions of masses of infinite extent, it might be shown that upon 
each series of orthogonal transversal surfaces, Chaslesian shells of 
infinite extent might be constructed. The level surfaces of these 
shells would be the orthooronal transversal surfaces of the same 
series, while their orthogonal transversal surfaces would be the level 
surfaces of the original Chaslesian shells and the other series of 
orthogonal transversal surfaces. 

166. To investigate the attraction of an ellipsoid upon an 
external point, it may be supposed to be divided into an infinite 



— 81 — 

series of elementary Chaslesian shells. Let then 

A x , A y , A z , be the semiaxes of the ellipsoid, 
a x , a,j, a z , those of the outer surface of either of the elementary 
Chaslesian shells, and let 

_ a x a y a, 

Ac A A' 

If, moreover, x, y, 0, are the coordinates of the attracted point, 

A x , A y , A z , are the semiaxes of the ellipsoid, which has the 
same foci with the given ellipsoid, and whose surface 
passes through the attracted point, 

a x , d y , d z , the semiaxes of the ellipsoidal surface, corre- 
sponding to the outer surface of the Chaslesian shell, 
and passing through the point of action, 

E 2 = A' X 2 — A x , and 

9 9. ?2 2 '9 A 9 9 

e*n = ct x — a x = a x — A x ir; 
the values of E and £ are the roots of the equations 



- - — I 

C2 ± 1 



x Al-\-E 



= 1js 



x a 2 +t 2 n 2 n 2 x Al-\-z 2 ~~~ 

The attraction of the Chaslesian shell upon the external point 
in the direction of the axis of x is by (77 2 ) 

. 7 aj.a u a z p' 2 da x . , a x a y a z p' 2 dn 

t: JL to JU T~r> 7 y • \k. J L fi U/ ~r~o 7 T~ • 5 

a x a y a z a x a x a u a z n 

in which the value of p is, by equations (76 9 _ 18 ), given in the 
form 



1 


= 




UL 


X 1 


/» 


D5 


,{xD x L)Y~ 




= 


5; 


X 2 

: (a 2 -|- s 2 ra 2 ) 2 
11 


-±2 

M 4 ■ 



(Al+ey 



— 82 — 
The differential of (81 2 i) after it is multiplied by n 2 is 

whence by (Sl 31 ) 

11 s 

(in = 77, c d c . 

p - 

This value reduces the attraction of the shell in the direction 
of the axis of x to 

, , a r a„a,n^ 7 ^ n A r A„A~d.s 2 

— 4 7i kx ,1 ,. , 'cat = — Inkx- 



The integral of this expression is the attraction of the whole 
ellipsoid. The limits of integration correspond to the values of a, 
for one of which the shell is evanescent, and for the other its surface 
coincides with the surface of the ellipsoid. But, when the shell 
vanishes, n is zero, and e is infinite ; and Avhen its outer surface 
coincides with that of the ellipsoid, n is unity, and e becomes E. 
Hence, the expression for the attraction of the ellipsoid in the 
direction of the axis of x is, if 

M is the mass of the ellipsoid, and 
./Tits mean density, 

ao 



DM 



?jMx r 

T7-2 



+ £ s)v/[(^ + £ *)(^ + 0(^ + £ 2 )] 



By advancing each letter in the series x, y, e, and x to the follow- 
ing, the corresponding expressions are obtained for the attractions 
in the directions of the other two axes. 
167. By the substitution of 



the equation (S2 24 ) becomes 

oo 

r>__3Mxr 
-"< K K 



168. By the substitution of 

A 



h x h%sim + Al-A${bl + Al-Al)y 



; , and 
6, 



u — — • 
x — A' ' 



the equation (83 3 ) becomes 



DM 



3 3Tx C hul 



3 M x r 

~ KTju. 



*Jl{A%-\-v.%{A\ — Al)}(Al-+ U l(Al — A'))] 



which formula, with transformations similar to the following, is 
given by Legendre. 

169. If A x is assumed to be the greatest of the semiaxes of the 
ellipsoid, and A z the least, let 

2 Al + e* 

i Ax-]-^' 

. 2 . Al — A% 

sm^ = -,T j|, 

Ai — Ay 

sin & = sin i sin cp , 

a - Al + E* 

sin = sin i sin C P ; 

and let the first and second forms of the elliptic integrals be 
expressed by the notation 



9^(jp — / sec$, 

J d> 

T 

&iCp = / cos$. 

J 6 



r 



— 84 — 

These equations give 

e 2 sin 2 9 — A 2 x cos 2 cp — Al, 
(A 2 y + e*)sm*(p = Al — Al-\-(Al — A 2 X ) sin 2 t P = (A 2 X — A 2 )cos 2 6, 
(Al + z 2 )sm 2 tp=Al — A 2 , 
(Al + z 2 ) sm 2 tp = (A 2 x — A 2 z ) cos 2 y; 

sin 2 (pd.a 2 = — (Al -f- a 2 )d.sm 2 (p = — (A 2 X — J. 2 )cosec 2 9<#.sin 2 <p, 
d.i 2 = — 2 (A 2 X — A 2 ,) cosec 3 cpcos(pdcp ; 

which, substituted in (82 24 ), reduce the expression for the attraction 
in the direction of the axis of x, when the ellipsoid is homogeneous, 
to the form 

Djl = SM.v f ; i ;> se ??a = 5*1 /%(.<) _ cos^)) 

= ,<* ^'. , .(9W'-M>); 

(A x — A*)*Bmh 
. f p _ 3M _ 3M 

the attraction is 

x sin 2 i v l l ' 

The same substitution gives the attractions parallel to the 
other axes in the forms 

D y S2 =Py i sin 2 9 sec 3 <3 , 

^ 



DJ2 =zPs f tan 2 g) seed. 



' 

But the differential of the logarithm of (83 21 ) is 

cotAZM = cote/), 
and, therefore, 

ZM = tan £ cot 9, 



— 85 — 

D, (tan (p cos d ) == sec 2 (p cos d — sin 2 (.1 sec (1 

z= secd(sec 2 cpcos 2 d — sin 2 d) 
= sec d sec 2 cp — sec d sin 2 d (sec 2 9 -j- 1 ) 
= secdsec 2 (p — seed sin 2 a'( tan 2 <p -]- sin 2 cp) 
= seed -|- cos 2 i sec d tan 2 9 — secdsin 2 d 
= cosd -J- cos 2 ? seed tan 2 9, 

_, , . . N _D, h (tan cos op) cos w cot ai sec 2 tan (9 — tan f) sin en 
2), ( sin if cos rp seed) = p . . — — = — — —r—. - 

9 x * ' ' sin i sin 1 

= cos 2 y sec 3 d — secdsin 2 cp 
= — cos 2 ?'sin 2 (/)sec 3 d -j- (1 — sin 2 c/)sin 2 ?')sec 3 d 
— sec d sin 2 cj) 

= — cos J «sm-'c/)sec d d -j- seed ( 1 — . /. I 
= — cos 2 ?'sin 2 cpsec 3 d -A- -^— ■ -(cos 2 d — cos 2 ?') 

' ' sin -% v y 

= — cos 2 ?'sin 2 <psec 3 d — cot 2 ? seed -j- cosec 2 /cosd. 
These equations reduce the attractions to the forms 
* 

_ ,-, -r. C rSCC 2 /cOS^ SCC# 9 • t-> / • a\1 

D,j & = Pi/ J . ! - sin 5 f- sec 1 D^ ( sin 9 cos 93 sec d ) J 

^ 
= P^/(^cosec 2 2?S i <f» — cosec 2 a'9^<£» — sec 2 ?sin c / J cos c f , sec0), 

D z £l = sec 2 iPz j [Z> 6 (tan cp cosd) — cosd] 
' 
= sec 2 ?*Pg(tan*cos0 — &i c P). 

170. The following values are derived from (83 23 - 2 4) and (81 15 ); 

2 , __ ^Tj + i? 2 __ ^_ 2 

C0S ^ — Jf+^s — 2T" 

• 2 rr> __ ^' — A\ __ A'* — A'* 
sm i _^_p-^_ _ f 

sin 2 0- -^" "^'' J ' 2 ' A? 



Ai+w 



f'2 J 



_ 8G — 

2 Q _ Al + E* __ A? 
cos v — Al ^_ Ei — A ,.„ 

. 2 ._ A\. — A*„ _ A'* — A'* 

Sm l — A%— Al~~A? — AT 

A'l A 2 An A'l 

cos 1 = a'i=a* = a?-a?- 

The equations of the attractions give, by means of these values, 
that of P and (81 15 ), 

sJ^=:22 x D x >-n = sec 2 /Psin<l>sec<£sec0(cos 2 — cos 2 *) 
3M , M 

4:71 



— A' x A' y A' z — —If 

This simple equation is due to Legendre, and the first of the 
two following equations which are obtained by the same process of 
reduction. 

A'; 2 D X & 9 ( A'z-n»Q\ ^ M os cB 

AlD r Si_2 2 (A 2 D«M\ — 3 -^- %.& in JE 2 — 

^x x —^-xk^x^—J—^Al — Al) * M'' 

171. By putting 

CO 

the attractions may assume the form 

D x £l=—ZMxD A -2 L, 

x 

Djl= — ZMzD A * L. 

in which the differentiations, relatively to A\, A 2 ,, and _4 2 , are 
performed without regard to the changes of E, dependent upon the 
formula (81 J5 ). 



— 87 — 

172. The equation (81 19 ) may, by means of the equations 
(84 2 _ T ) be written in the form 

or by the substitution of the value of from (83 24 ), 

x*-\-z*sec 2 ® ■ y 2 1 

il I 



A% — A% ' Al cos 2 (p-\-Al sin 2 (D — A\ sin 2 0>" 

173. When the attracted point is upon the surface of the ellipsoid, E 
vanishes, and the value of becomes 

COS<P:=-^. 

174. When the attracted point is within the ellipsoid, the Newtonian 
shell, of which the outer surface is that of the ellipsoid, and the 
inner surface passes through the point, exerts no action upon the 
point, and the attraction is reduced to that of an ellipsoid similar to the 
given ellipsoid, and of which the surf ace passes through the attracted point. 

175. When the density of the ellipsoid varies in its interior, 
in such a way that each of its component Chaslesian shells is homo- 
geneous, It is a function of e, and after its substitution (82 24 ) may be 
integrated. 

176. When the ellipsoid is a homogeneous oblate ellipsoid of revolution 
the various formulae become 

A^ = Ay, 



i=$ = 



x 



+ «?/ 2 4-s- 2 + s 2 tan 2 ^ = (^l — ^)(l-f cot 2 <£) ; 



s 2 tan 4 <£ -f (r 2 — A\ -f .A 2 ) tan 2 * = A\ — A\ ; 
D x il = Px C sin 2 ? = \Px{2 <P — sin 2 <P) , 

* 

$ 
D y il = Pg f sin 2 g) = |P^(2<£ — sin2<£), 

U 6 



88 



DJ2 = Pz ftanhp = P.e (tan* — *) . 

' 

177. FFAera //^ ellipsoid is an homogeneous prolate ellipsoid of revolu- 
tion, the formulae become 



A = 


= A*. 


i- 


-\n 


cp = 


= A, 



a? »+y»-+ a « + (y" + »)tan»*=(^ — -A»)(l + cot»*), 
(y 2 + z 1 ) tan 4 * + (r 2 — 4» — 4") tan 2 * = A\ — A\ 



I 2 - 



2^X2 = Pa; / sin 2 g)sec9 

^ 

= Px [log tan (-1 rr -f- i *) — sin *] , 

Dyfl = Py / sin 2 ^ sec 3 9 

o ^ 
= |-P# [sin*sec 2 * — logtan (|- tt -j- | *)] > 

■Z/^iO = Ps J sin 2 (j)sec 3 g) 



= iPg[sin*sec 2 * — logtan {{n + |-*)]. 

ATTRACTION OF A SPHEROID. LEGENDRE's AND LAPLACE'S FUNCTIONS. 

178. The investigation of the attraction of a spheroid is 
greatly facilitated by the introduction of certain functions which 
were first conceived and investigated by Legendre, but which 
became so fruitful in their more general form, given in the subse- 
quent researches of Laplace, that they are usually designated by 
the name of the latter geometer. A method will be pursued in 
their development and discussion which is similar in some respects 
to that given by Jacobi. 



— 80 — 
170. Let 

11= cos (p -f- asm cp cost], 

and if any power of II, denoted by n, is developed in a series of 
terms arranged according to the cosines of the multiples of i], let 
any one of the terms be denoted by 

in which \_m~\ denotes the number of accents of C P. The required 
power has then the form 

II' 1 = 2 m (i m <P™ cos m i] ) . 

00 

180. The value of i/"is not changed by reversing the sign of 
i], and, therefore, the series remains unchanged by this reversal of 
sign, which gives 

or <P\~ m i = ± &w = (— 1 )'» <p™ ; 

in which the upper sign corresponds to the even values of m, and 
the lower sign to the odd values of m. The equation (89 n ) may 
also be written 

H n = <P n -\- 22 m (i m <P™cosmr)). 

181. The integral of the product of (8922) by cosmi] is, by a 
well known theorem 

f (H n cosmi]) =2 71 i m <P [ ™\ 
v 

The derivative of this equation, relatively to (p, reduced by the 

condition 

D^ H= — sin (p -\- /cosy cosi; , 

12 



— 90 — 
becomes 

2- 



2ni m D lk <P [ ™' ] = n I [_H n ~ 1 cosmi] ( — sine/) -\- icoscpcosi])'] 
U n 

o 

2,r 

= 11 J [jST" _1 ( — sm(pcosmi]-\-^icos(p(cos(m-\-l)i]-\-cos(m — I) 1 ]))] ; 



whence, by (89 27 ), 

D^ <P™ = — n sin qp *Wj 4- \ n cos <p ( *[;"__! 1] — *£!^ 1] ) . 

182. The derivative of (SO^), relatively to 1], reduced by the 
condition 

D H= — z'sin (p sin ij 
becomes 

oo 

in H n ~ 1 sin ip sin i] = 2 2l 7!1 (in i m [ ^ sin in i] ) . 

i 

The integral of the product of this equation by smmi] is 

2tt 

in sin <p j ( H m ~ l sin i] sin m -jj ) == 2 tt m / OT «£»£* ] , 







or 

6 

which becomes by (89 27 ) 



in 9 f [#" OT - a (-| cos (m — 1 ) i; — £ cos (m -\- 1) ij )] = 2 re m i m <P™ 



183. The equation (89 2T ) may assume the form 

J[_II n ~ J (cos(pjcosm , )]-\- : kismcp(cos(m-\-l)i]-\-(cos(m — l))~\=2ni m <P [ ™\ 
V 


which, reduced by (89 27 ), gives 

<pw = cosy <££*-! i + I sine/) ( £fc 1] — *L'- + i 1] )- 



— 91 — 

184. The remainder, if (90 8 ) is subtracted from the product 
of (90 24 ), multiplied by cotcp, is 

rf/»>~l 

= sm m + 1 cpl) c0 ,,-^-. 

' C0B V sin '" cp 

The sum of (90 21 ) and n times (90 31 ) is 

ncosqp $£*!.! + wsinc/)c£»[r-l 1] = (n -j- m) c P\l" ] , 

the first member of which becomes identical with that of the 
previous equation, when m is increased by unity. Hence, 

*f;" +1: _ 1 n 0™ sin cp n Of** 



sin '" cp n -)- m -)- 1 * sin '" cp n -\- m -\- 1 cos sin '" qo ' 

or, if m is diminished by unity 



sin "' (p n-\- m cos ^ sin m ~ 1 cp ' 

If the sign of m is reversed in this equation, it becomes by 
(89 17 ) 

ain-g>#M = ; ^Z>„ # (ain"+ 1 g>#£ ,1 + 1 i). 

185. It will be found convenient here and elsewhere to adopt 

the functional notation 

i 

rh=f x {-\o g .z)\ 



which gives, by a familiar formula, or by simple integration by 
parts, when h is positive, and k an integer, which is less than h -f- 1, 

i 
rh = h(h — 1)(A — 2) (A — A -J-l)£(_ log*)*-* 



= A(A — 1)(A — 2) (A--A-j-l)r(A-^A), 



— 92 — 

and 

*(*_1)(*_2)....(*_* + 1) = J ^. 

When h is an integer, and k the next smaller integer, this 
formula becomes 

1.2.3 h = rh. 

With this notation, Taylor's theorem assumes the form 



<& 



186. The equations (91 16 ) and (91 20 ) give, by successive sub- 
stitutions in each other, and the use of the preceding notation, 

sin" ! <p cos ^ sin m > i^'T'V^ sin'"' 9' 

(— 1)"IT(m — w)sin m «p*£" ] = (— 1)T(« — w')£^ m (siiT'g>#£ ,, ' ] ) 5 

in which negative differentiation must be interpreted to be integra- 
tion ; in the former equation, when n is negative, in' — | — ?z — | — 1 and 
m -j- w -j- 1 must be positive ; while, in the latter equation, n, 
n — m -\- 1, and n — in' -\- 1 must all be positive. When n is posi- 
tive, but 11 — m -\- 1, and n — in' -f- 1 are negative, the equation to 
be substituted for (92 15 ) is 

$m m cpQ>™ _ D'^ m (sin'"' cf^;"' 1 ) 
I\m — n — l) ~ l\m' — n — l) ' 

which equation is also to be used when n is negative. When n and 
11 — m-\-l are positive, but n — in -j- 1 is negative, the combina- 
tion of (92 15 ) and (92 23 ) gives, by representing by ii', the greatest 
integer contained in n -\- 1, 

(_ 1) «'-»r(n — m) s m m q,cp^ />' — n — 1) Jg^" (sm^xpOT' 1 ) 

I\n — n') I\m' — n — l) 

When 11, 11 -j- mf -f~ 1, and m -\-m-\-l are all negative, the 



— 93 



equation to be substituted for (92 13 ) is 






r(—l—n — m)sm'"q) l\—l—n — m') cos sin'"' cp ' 

When n and n -J- »*' -|- 1 are negative, but m — (— « — |— 1 is posi- 
tive, the combination of (92 13 ) and (93 3 ) gives, by representing by 
n r , the greatest integer contained in — 1 — n, 

r(m-\-n)0»i (_l)»'-»T(-l-n-»i') / , ffl _ B , 0^ 



r(n-\-n')sm m cp I\—l — n — m') cos sin'"V 

There are peculiar considerations which simplify the investiga- 
tions, when 11 is integral, whether it be positive or negative ; and 
these are the cases to which most of the subsequent investigations 
are limited. 

187. By reducing m or m r to zero, the equations of the pre- 
ceding section give, for positive values of n, 

<j*" ] = rv F, \ N sin"? Z?" <fi» = (— IV*— T \ . m f m <P n 

F(m-\-n) J cos^ » v j r( /l — ?ii)sm'"(pJ C0S( j ) " 

, -.w Fnr(m — n — l) P>» , 

\ x ) r(n — „')/*(»' — » — l) sin»q) J cos <p "' 

and for negative values of n 

$W, /> -* -- 1) f« ^ 

J-(— 1— n)sin>Jcos0 n 
( — 1)'".T( — 1 — ?t — m) . m j- )m * 

~ r{m + n) /'(— 1 — n) Sm ^ cos * * » ' 

188. IFifo?? 7? is zero, it is easily seen that 

K°=1 = <P , 

and that, for all other values of m 



— 94 — 

189. When n is a positive integer, and h is also a positive integer, 
the equation (91 8 ) gives 

(272 -(- h) <P% + h] — n cos (p <PH l ±i ] + n sin cp <P%Zi + /,] . 

If, then, the terms of the second member vanish for any value 
of n, they will also vanish for the next higher value of n. But they 
vanish, by the preceding section, when n is zero, and, therefore, they 
vanish for every positive integral value of n ; that is, 

Cbfr + h] A 

M n \J . 

or the series is finite for positive integral values of n, and contains onlg 
ii -\- 1 terms. 

190. The substitution of the preceding equation in (91 8 ) gives 

<£M = i sin cp <&f-p = ( I- sin tp ) m <P\?Z™ ] 
= ( \ sin cp Y O = — sin" cp . 

which equation, substituted in (93 16 ), gives 

®n = { ^f A"os^( *lT ] sin"9) = ^jrD'U{— sin »" ; 

rf,[»»] — . sm " l(p 7}» + ™ c s i n 2 fD v i 

r n — 2-r(m + n) cos ^ ^ ^ 

— 2T(n — m)sin m <p cos 4> ^ bm ^ ' 

£y 20/^ ^Ae coefficients of the development are obtained when n is a positive 
integer. 

191. When n is the negative of unity, the equation (89 27 ) gives 



o 



But the value of II gives 



— 95 _ 

1 1 cos cp — i sin qp cos jj 



II cos cf) -\-i sin cp cos// cos a qp -j— sin 2 gi cos - ^ 

i sin cp cos t] ■ cos qp 

1 — sin 2 qp sin 2 rj *^ cos 2 /; -\- cos 2 qp sin 2 /; 

^ i sin cp cos ij \ i sin ft cos i\ , cos qr D v tan ?/ 

1 -(- sin qp sin j? 1 — sin qp sin // ' 1 -|- cos 2 ft tan 2 j/ ' 

the integral of which is 

/ 77 = — 4- 2 log -J- sin <y> sin q , ^ t _ i] / CQg ^ ftn \ 
J ?/ // - 1 °1 — sin ft sin ?/ ' v ' ' 

Hence, by passing to the limits 

2n0_ 1 = 2 n, 

When « and ra vanish, equation (90 31 ) becomes 
<P = cos <p <&_ ! — sin <p <P'_ j , 



whence 



r /■ 1 COS ft . -, 

C P, = : = — tan i (o , 

sin ft ^ ' 



Equation (92 13 ) gives, then, 

AM — sin '"^ 7> m - 1 sP0 2l m 

**-i — 2J'(m-l) cos r eC 2 ^' 

192. When n is any negative integer, it is more convenient to 
write the formulas with the sign of n reversed. With this change, 
the sum of the product of (90 31 ) multiplied by ( — nsiny), that 
of (90 8 ) multiplied by cosy, and that of (90 24 ) multiplied by 
( — cosecy) becomes 

«<£^ ( t+u = cosy DA*™\ — («shi<p -f -¥-) <P [ ™1 

' \ ol 11 u / 

sin»- i ft n «ea / ff) -,, . , B+i-»\*h] 

= -^ -v-s=i ( ( 2 n — 1 ) sm w -\ -. ) <Ptl" t ; 

sec ft 9 sin " 1 cp V ' ' ' sin ft / 

which, when 

m = n — 1 , 



— 96 — 
is reduced to 

The successive substitution of 1, 2, 3, &c., for n, gives, by means 
of (91 12 ) 

r sin " A <jp 

T r„l 2« 1 . -rr,, 11 2n(2ll 1) . -,r„ ,-, 

*^ ] ( „ + l) = T~ Sm ( P *-* = 2^T- sm 9 *-^ ] 






2 " (/'») : 
The substitution of this value in (93 2 i) gives, by (94 20 ), 

*-<.+u = 7^^W( sin > clJ -Wa)) = 2^^cos^(— sin»"= <P n ; 
and, therefore, for all values of m less than n -J- 1, 

/ -I \m r(n — m)r(n + m) ~ [m] 

v > {my n ' 

The equation (91 8 ) gives, when m — n vanishes, by (96 ), 

&-V4d = — cot9>*w„ +1) =2^- 2 (— BnyJ-^cosy; 
whence, by (90 24 ) 

*^Sii, = — 2cosecy *w. _ <*>[»- ri ]} . 

From this equation the successive values of c P l l ] n may be deter- 
mined by successive substitution of 1, 2, 3, &c, for n, or they may 
be determined by the equation derived from (91 20 ) and (96 9 ), 

cos<4 

<£[» + !] _(2n + l)r(2») 1 f , . , y 



— 97 — 

The remaining coefficients, in which m is greater than n, are 
then to be determined by the equation derived from (92 13 ) ; 

<£[»] _ sin '"9 rm-n ®-l 
-• r(m — m) cos s in>" 

193. In order to apply the preceding investigations to the 
problem of attraction, it is requisite to introduce the form of polar 
coordinates, of which zenith distance and azimuth is the familiar 
instance. For this purpose let the following notation be adopted : 

(jfyis the angle which a line /makes with the axis, 
& f is the angle which a plane, drawn through the axis, parallel 
to/, makes with the primitive plane. 

The distance /, between two points, of which the radii vectores 
are r and o, is given by the equation 

/ 2 = r 2 -f- o 2 — 2ro(cos</vcos(^-(- sin 9V sin 9) cos (t\. — 6 )) 

— r 1 9 — 2 or cos . 

Hence the notation 

i=\/— 1, 
H f = cos(p f -f- ism (p f cos (i] — 3 f ) , 



gives 



+fHf = +/cos (f f + ^/sin 9ycos (t] — & f ) 

= rcos(f r — ()cos(p p -f-^cos?j(rsin9vcos$ r — osinro cos^ ) 
-j- ismi] (rsm(p r sm& r — osiny sint) ) 
= rH r -qII p ; 

in which the upper sign is to be used when r is greater than 0, and 
the lower sign when r is less than 0. 
But it follows, from § 191, that 

/ *27tJ v fH - r 27tJ ri rir r — Qir p 

' 

13 



— 98 — 



2tt 



o o ' 

in which the upper sign corresponds to r, greater than o, and the 
lower sign to r, less than q, and the series represented by the fourth 
member corresponds to the former of these two cases, while the 
series represented by the last member corresponds to the latter 
case. 

194. If, in the development of the preceding series, 

Q n is the coefficient of -^+y, and 

Q' n is that of ~ ; 

the series become 

The values of the coefficient in these series are determined by 
the equations 

o 

2tt 

V* — 27tJ v Ep im 

o r 

If the additional notation is adopted, corresponding to (89 7 ) 

CO 

II; = [ r ] m -J- 2 2 m {i m [r]M cosm (rj — 6,)) , 
the values of the coefficients become 

Qn = M- M-e+u + 2i„((- l) w M? 1 [r]?). + i,cosm(fl p - d,)), 
Q: = H. M-c+i, + 2 ^ OT ((- 1)* [r]W M?UiCosm(fl p - 4,)). 



— 99 — 
Hence, by (96 18 ) and (93 16 ), 

Q n = Q:= M.[g].+2f w ( r( "~" / ) y y , " h,,) MS" 1 MS* 1 cob« (d p - <),)) 
=H.M,+2i,(J ^^ 

195. The equation (45 8 ) gives for the value of the potential, 

Hence, by the notation 

U n =f{lcQ n Q«), ' 

ir - f *« 

the potential becomes 

co 7-r oo 

X> = 2 — - ■= ,2 C Z7V) . 

o ? o 

"With the notation of (58 23 _ 27 ) and (97i ) these values become 
do = Q 2 di}'d() = () 2 smcp d(fpd&pd(>, 

The first form of 12 in (99 17 ) is to be used for all values of o 
less than r, and the second form for all values of (> greater than r. 

If, then, Jc is supposed to vanish for all points of space in which 
there is no attracting mass, the limits of integration for the value of 
TJ n must include all the attracting mass for which () is less than r, 
while those for U' n must include all the attracting mass for which o 
is greater than r. 



— 100 — 

196. By substituting in (99 2 i_ 23 ) the values of Q n given in 
(99 2 ) the resulting values of U n and U' n have the same form with 
Q n so far as the elements of the direction of r are involved ; so that 
the value of the term of U n which depends upon the angle m& r has 
the form 

[r]L'" ] (AH" ] cosm6 r -j- B [ ™hmmA r ), 

in which A 1 ,'"'* and B { ™ ] are independent of the form of the body, 
and the number of such constants included in the most general 
value of U n is 2 n -f- 1. 

197. It is expedient to introduce, at this point of the discus- 
sion, some important properties of Legendre's functions. The 
following theorem, given by Poisson, is of especial use in facilitating 
their investigation. 

If JV P denotes any function of the elements of direction of q, and if, 
after the performance of the integration expressed in the second member of 
the following equation, Q is made equal to r, ivhich condition is intended to be 
denoted by the subsequent parenthesis, the second member ivill be reduced to 
the first member, that is, 

N ' = i-«) f /• [? = '•]' 

' 

To demonstrate this theorem, it is to be observed that all the 
elements of the integral vanish, except those for which 

f=0, 
that is, for which 

1' r> 



r = Q, Q 



If, then, 



i] denotes the angle which the plane of r {) makes with any 
assumed fixed plane passing through r, 



— 101 — 
the integral becomes, by (97i 6 ), 

J_ f r(r 2 — Q *)X P _Xr 2 f p(V 2 — g 2 )sin'p 

&■ = *] = #. 




JV; r 2 — e 2 

"2" <>(»•-<?) 



198. The equation (97i 6 ) gives, by means of the first form of 

(9S I5 ), 

=/«i.((2n + l)C.^i), 



which substituted in (100 20 ) reduces it to 

u 

by adopting the notation 



*=f.(n£l(«-*>) = f^ 



o r 



in which it must be observed that when « is zero it must be 
retained in the written expression to avoid confusion. It may also 
be remarked that, from the comparison of the forms of (99 12 ) and 
(101 2 i), the most general form of N^ is the same with that given in § 196 
for U n . 

199. If the given function is such that, for every value of n' 
different from n 



— 102 — 
the equations (101 17 _ 2 i) give 



o r 



(«.^P)=i#-i-^. 



The theorems expressed in the last two equations are of fundamental 
importance, and tuere given by Laplace. 

200. The theorem (102 7 ), not being limited to any special 
direction of r is true for all directions ; and, therefore, the most 
general form may be substituted for Q n , which can be obtained by 
combining all its special values in any linear function. Any such 
general form would be the same with that of JYjf\ and if it is 
denoted for distinction by M l p n/] , the theorem (102 7 ) assumes the more 
general form given by Laplace, 

in 

f (MjplVF) = 0. 

r 

201. In considering the attraction of a spheroid upon an external 
point, ivhich is so remote that r is greater than any value of q let 

u be the value of q for the surface of the spheroid, and 

J P 



the function which is denoted by the second member of this equa- 
tion being developed in the form of a series of terms of Legendre's 
functions, by means of (101 2 i). The equation (99 21 ) becomes, by 



— 103 — 
means of (102 5 ), 

and the potential is 

o "V(2«+l)^ +1 rn )' 

202. If the point is so remote that the squares of the linear 
dimensions of the body may be neglected in comparison with the 
square of the distance of the attracted point, it has been shown in 
§ 128 that the attraction is the same as if the body were condensed 
upon its centre of gravity. In this case, therefore, if the origin is 
assumed to be the centre of gravity, the potential becomes, as in 
(56*), 

fl = = = i(0l + f). 

In all cases, then, in which the origin is the centre of gravity, this equa- 
tion gives for an external point which is so remote that r is greater than q, 

U = m, 
0i=O, 

r l g n \(2?i-\-l)r n + 1 ""»*/■ 

203. A homogeneous ellipsoid can always he found, of which the 
potential, for any external point, developed in the form (103 20 ), will be iden- 
tical with this expression in its tivo first terms. To demonstrate this 
proposition, and develop the mode of investigating the ellipsoid in a 
given case, it may be observed that, if the centre of the ellipsoid 
coincides with the centre of gravity of the given spheroid, and if 
the mass of the ellipsoid is the same with that of the spheroid, the 
potential of the ellipsoid, for an external point, has the form (103 20 ) 
with the same first term. The difficulty of the demonstration and 
investigation is thus reduced to the consideration of the second 



— 104 — 
term. The general form of this term is, by (99 4 ) and (100 6 ), 

R { P = A(cos 2 (p r — i) -f- sin cp r cos (p r [B cos $ r -\- .S'sin^) 

-f sin 2 (f r ( Ccos 2& r -\-C sin 2 1\) 

= — ±A -f J[cos 2 ^ -f <?(cos 2 ^ — cos 2 j) 

-\- i> COS^COS^ -J- .# COS.COS -f- 2 6 T COS^COSy 

= -S x (^ (cos 2 ^ — -1) + ^cosj cosy ; 

in which last form the arbitrary constants C x , C y , C z , B x , B y , and 
B z are introduced, for the sake of symmetry, and in which the six 
constants are only equal to five, by reason of the equation 

In the especial case of Q 2 , these constants become, for the axis 
of x, 

B x = 3cos^cos^, 

V x 2" COS x J 

and similarly for the axes of y and z. 

The equation (76 6 ) of the homogeneous ellipsoid, of which the 
axes are the given rectangular axes, gives, for the surface of this 
ellipsoid, 

1 „ cos 2 £ 

If, therefore, K is the density of the ellipsoid, the equation 
(102 27 ) becomes, in this case, 

u 

o J f J s » + 3 



— 105 — 
and hence, by (103 2 ), 

v 

To obtain the value of U 2 , it must be observed that, by (104 17 ), 

4tt 

/V#)=o, 



■<l> 

r 

because, from the symmetrical form of the ellipsoid, the value of u 
is not altered by changing either of the angles upon which it 
depends into its supplement, while the sign of B r x is reversed. 

The remaining terms of the integral contained in (105 2 ) have 
the form 

4 7T 4 7T 47T 

f ( U 5 C' Z ) = f f (« 5 COS 2 § = ff (u 5 COB 2 cp p ) . 

^ ^ ^ 

But, by the equations 

cos^ = sin (p cos & p , 



(104 25 ) becomes 



cos" = smcp n sin£„, 

y >t i> 



1 _ cos 2 (jfp . sin 2 cf'p cos 2 dp , sin 2 rjr^ sin 2 0p 

cos% i sin 2 fy , I 1 cos 2 fl p sin2g p \^^ f ,2„ 

= « -j- bcos 2 (p , 
by putting 

cos 2 p . sin 2 fl p , / 1 , 1 \ , - / 1 1 \ , 

= W + fl'cos2d p ); 

14 



— 106 — 
l | l 

y __ J J_ 

h = -7-, — «. 



The integral (105 15 ) is, therefore, 

i~ 2 7T 7T 

|J (tt B cos a 9>p)=fjT 5 J (u 5 cos> p siny p ) 



r 



Bat 



3 f f cos 2 qrpsingr p 
2 J fy J <jp p (a + &cos 2 <jp p )2 " 



J* cos 2 gip sin cpp — cos 3 qp 

fp ( a ~f" 6cos 2 </p) 2 " 3a(a-|- 6cos 2 g>p) 2 ' 



cos ' qpp sin g-p 



2 2J^ 



r - 

J gy> (a -)- 6 cos 2 qrp) 2 3a(a-[~^) 7 ^ a ' 

r i=r i-=r * 

J# p a J26p2a J 2 a' + b' cos 20 p 



P 

tan [ p 

y/ (a! 2 — 6' 2 ) y/ (a' 2 — 6' 2 ) 



= ^4tant- 1 (^tan^ p ) : 

2tt 



These values, with that of the mass of the ellipsoid, reduce 
(106 8 ) and (105 2 ) to 

| / (« 5 cos 2 f/) p ) = 2nA x A y Al, 

r 



— 107 — 

If the axes of the ellipsoid are not those of x,y, z, but oix r ,y',z f , 
this expression, by means of the equation, 

r r x i , r — 'y . r z 

COS 2/= COS x COS x > -f" C0S y C0! V~r C0S z C0! V> 

becomes 

U 2 = ^mZ x [c' x '(cos^ : — i) + ^cosjcosl]; 

in which 

C = ^(^ r cos 2 ^), 

^ = 2^(^^cosJcos^). 
This value becomes identical, therefore, with that of (104 8 ), if 

n" -0 p 

B" = —B . 

x 3m x 

204. If the potential and its component functions for the 
ellipsoid are denoted by the letter e written beneath them, the 
potential of the spheroid for an external point, for ivhich r is greater than o, 
becomes 

e 3 e 

205. A transformation of coordinates, which is the reverse of 
that by which the equations (107 17 ) were obtained from the reference 
of the ellipsoid to the axes of the spheroid, would bring the equa- 
tion (104 19 ) to the form (107 16 ). From the forms of the expression it 
is obvious that this transformation is identical with that by which 
the general equation of the second degree in space is referred to the 



— 108 — 

axes of the surface. Hence, if S x ,, S y ,, and S z ,, are the three roots of 
the equation 

( a:— 8) ( c;;— s) ( c:—s) + 2 &b;b:— z x \bi\ <%— #)] = 0, 

they are the squares of the semiaxes of the ellipsoid. But it must 
be observed that the mass of the ellipsoid, being the same with that 
of the spheroid, gives the equation 



Cf Cf Cf J 3m V 



The condition (104 14 ), however, shows that the values of C x , G y , 
and C z , in (104 8 ), may be increased or decreased by the same quan- 
tity, without changing the value of (104 8 ). The values of C", C' y ', 
and C" may, in like manner, be increased or decreased by the same 
quantity, which change will produce the opposite effect upon the 
roots of (IO83), until, at length, they may satisfy the equation (108 9 ). 
This common increase or decrease of all the roots of (108 3 ) corre- 
sponds to the performance of the same operation upon the squares of 
the semiaxes of the ellipsoid, that is, to a change of the ellipsoid, 
given by (108 3 ) into another ellipsoid, which has the same foci and 
the required mass. The change of mass is, however, more simply 
accomplished by an increase or decrease of the density of the ellip- 
soid ; and, in this view of the case, it is requisite that the value of 
the density be determined by equation (108 9 ). 

206. If the point is without the spheroid, but near its surface, 
it is generally necessary to combine the forms of the potential given 
in (99 16 ). Thus, with the notation 



/ = the integral for all directions of u greater than r, 

/= the integral for all directions of u less than r, 




— 109 — 
whence 

u ij) u ip u ip 
the value of the potential may be expressed in the form 

S2 — 3: (— -I- V r n \ 
in which 

v Y 



u 
J ib UP Q 



Bat it may be observed that, by (109 2 ), 

4 77 « 



' 

whence, by putting 



\J ihJp U ib J P Jib J p' " S 

Y ' r ' 



u u 



and using ZZ„ in the signification of (99 12 ), the potential assumes the 
form 



<2=f(^-jy,„ <?.)). 



207. A similar investigation may be extended to the ellipsoid 
of § 204, and if 

SI' is the value of LI of (107 24 ) 



— 110 — 

the value of the potential for a 'point which is near the surface of the spheroid 
may assume the form 

£2 = n>-Z n f((V pn -V pn )Q n ). 

«/ 1p e 

208. If the form of the spheroid differs but little from an ellipsoid 
ivhich has the same foci tvith the preceding ellipsoid, and if it has a constant 
density for all that portion for ivhich Q is greater than r, a combination of 
two homogeneous ellipsoids may be substituted for the single ellipsoid, both of 
which have the same foci, ivhile one coincides very nearly with the spheroid in 
form and density throughout the portion exterior to r ; and the other, being 
much smaller, has the requisite positive or negative density to give the alge- 
braic sum of the masses of the two ellipsoids equal to that of the spheroid. 
The combination of the two ellipsoids upon any external point is 
the same with that of the single ellipsoid, and the larger of the two 
may be substituted for it in the values of Fin (110 8 ). 

If, in determining the values of Ffor the spheroid or the ellip- 
soid from (109 22 ), u is supposed, for every direction in which the solid 
is contained within the sphere, of which radius is r, not to refer to 
the surface of the solid, but to coincide with r, the value of V van- 
ishes for any such direction, and it becomes a continuous function, 
of which the derivatives are discontinuous. The equation (101 22 ) is 
applicable to such a function, for the argument by which it was 
established was independent of this condition. With this modifica- 
tion, therefore, the accent may be omitted in the integral sign of 
(109 27 ) or (110 3 ), and the limits of integration extended to every 
possible direction, and the result may be simplified by means of 
(101«). 

In the present case, in which Jc is constant, equation (IO922) 
becomes 

Tr k / M » + 3_ r » + 3x jfc r r n -, 



— Ill — 

whence 

If, then, it is assumed that 

0p = u — r, 

z' =u — r, 

e 

the binomial theorem gives 



and if n is changed into — (n -j- 1) 

zi. (w «-> _ r «-) = -. i, ( ffi ~ * 1+ m) (- r)-p»-') g ?) . 

n — 2 v y 1 m \ I(n — 2) J'.m v ' 9 ) 

These values, substituted in (111 2 ), give 
v v £ f * / J> + 2) (-!)'- j>-3+m) \ -. 

' P J P" — f 4 J^r— » Vi> + 3 — m) I r(n — 2) A P p 'J 

= (2, + l)/4H^-< 2 )+i(^-^)+ ? 42^^-</)] 

,£ r__* / /'("+2) i ( -i) w n»-3+"») \ r ^ vmNi 

^ _ 7"'L/ , ^r'"- 1 Vr(w + 3 — m) ' J'(» — 2) / V "> P>\' 

This value may be substituted in (110 3 ), and the result reduced 
by means of (101^). 

209. If the spheroid is not very different from a sphere, and if the 
difference in form hetiveen it and the larger of the tivo combined ellipsoids is 
so small that, in consideration of the large divisors, the terms of (111 20 ) may 
he neglected, in which m is greater than 3, (111 20 ) is reduced to 

V 9 n— V pn = {2n+l)W p , 

e 

if 



— 112 — 

and, by (110 20 ), 

W r =0. 

But the value of the potential, derived from (110 3 ), becomes in 
this case by (111 28 ) and (101 22 ), 

& = & - Zn ((2» + 1) J ( W p Q n )) 

= S2T — ±n2 n W™ = n' — w r 



so that, in this case, the form (107 24 ) is applicable to every external point. 
This conclusion, and the mode of investigation, includes Poisson's 
analysis of the spheroid, which differs but little from a sphere by 
which it was suggested. 

210. The formula (107 24 ) gives, for the attraction in the direc- 
tion of the radius vector, the expression 

D r a = DJ2 + l n [(» + 1) ( U n — Z7„)r-<»+ 2 >] . 

e 3 e 

Hence, the equation is obtained 
D r il + li2 = D t n +li2 + ii.[(2n + 1) ( U m - R.) *-<"+«], 

*' e * r e 3 e 

which, by (103 2 ), is reduced to 

Z> r I2 + li2 = D r £2 + 1/2 + 2^ [(i?w _ ££]),.-<•+«] . 
or 

A[^r(i2 — i2)] = 27rv/r5 n [(iZpP — ^^)r-(»+ 2 ']. 

e 3 e 

211. If the spheroid is homogeneous, having the same density 
with the ellipsoid, the equation (104 31 ) gives 



113 — 



R . — - n n + 3 

e ' » -J- 3 e K 



Bj assuming, then, the values 

U T Zp M.j U r Zp = U p: 

' e e 

tt",, — ^ n + 3 *»+ 3N i f - - n *.»+3\ 

V — „_|_ 3 W fp >> K ' n — n + 3^ 1 t r j ' 

12' = LI — & ; 

the equation (112 28 ) becomes 

^i2' + ii2'=_2 Jt A-„,f,[n'?0)* +2 ]. 

212. If the attracted point is upon the surface of the spheroid, the 
preceding equation becomes, if 12 is the potential at the surface of 
the spheroid, 

1 



' r 3 

213. If the spheroid differs so little from the ellipsoid that the square 
of the distance between the surfaces of these tivo solids may he neglected, the 
notation 

y^z — s, 

e 

gives 

■> — g P 1/p — 1p Up- 

214. If, moreover, the ellipsoid differs so little from a concentric 
sphere, that the product of the difference between the radius vector 
of the ellipsoid and the radius of sphere, multiplied by the distance 
between the surfaces of the ellipsoid and the spheroid, may be neg- 
lected, the preceding equation is reduced to 

V P «=y P ; 
15 



— 114 
and (113 17 ) becomes 



AT 



In this last form, the sum of the terms in the second member is 
extended to include the whole series, because the first terms which 
vanish in the exact formula, may become sensible in the approxi- 
mate form. But, 

oo 

ifr — " n <y r ) 



and, therefore, if E is the radius of the sphere, 

215. If, again, 

£2 Q = the potential of the ellipsoid at its surface, 

e 

£2' = the potential of the ellipsoid at the surface of the 

e 

spheroid, 

O" CV £2 . 

^-0 — "-o — ^0 } 

e e e 

I 

the general equations 

CO 

e e 



give 



n';=-ini„(^Lmr-^<,).. 



D,a 



ir 



r"'-0 



^.i^m^^'-^)- 



Since the second members of these equations are multiplied by 



— 115 — 

y„ the values of the other factors may be reduced to those which 
belong to the sphere. Hence, B p » becomes a constant quantity, 
and, therefore, 

for all values of n except zero, in which case, 

7?i — -/? 3 -- 7? [01 - 
and the above values become 

e 

D r QZ = \itKRy w 

e 

which give 

The sum of this equation and (ll^) is, by (113 9 ) and (114 20 ), 
D r S2 + ±<2 = DJi + ±n t . 

216. If the ellipsoid is itself the sphere, the equation (58 8 ) gives 

e 

D r tt = — ±7tKfi, 

e 

which, substituted in (115 n ), gives 

This is the equation given by Laplace for a spheroid which 
differs but little from a sphere, and is the fundamental theorem of 
his investigations upon this subject. 



— 116 — 

217. If the attracted point is ivithin the spheroid, and at such a dis- 
tance from the surface that r is less than the value of u, the formula for 
the potential is, by § 195. 



12 — jJ (-^- 4- U'iA 



in which 



* 

J ib J P Q 



It may also be shown by the method of §§ 208-209, that this 
same formula is applicable, if the point is quite near the surface, and if the 
spheroid differs so little from a sphere that the square of the difference may 
be neglected. 

218. The important discussions in regard to the convergency 
of the series, derived from Legendre's functions, are deferred, on 
account of their great length, to the volumes which will be devoted 
to the application of the Analytic Mechanics. 



IV. 

ELASTICITY. 

219. The laws by which the elementary forces of cohesion and 
affinity vary with the mutual distance and direction of the particles 
and atoms are undetermined ; and, therefore, the delicate inquiries 
involved in the constitution and crystallization of bodies are not yet 
subject to the control of geometry. But it is sufficiently apparent 
that these forces are insensible at sensible distances, and that there are 
peculiar laws of mechanical action corresponding to the three states 



— 117 — 

of (/asses, liquids, and solids. The peculiarity of these states consists, 
principally, in the facility with which the particles can be moved 
relatively to each other, and in the phenomena which arise from 
such motion, but especially in those of the disruption of solid bodies. 
As long, however, as the relative positions of the particles are so 
little disturbed that they return to their initial state when the dis- 
turbing cause is removed, the precise law of molecular action is not 
required for the investigation of the small changes which the consti- 
tution of the body undergoes, and which are treated as phenomena 
of elasticity. 

220. To analyze the changes of form of a system of material 
points which constitute a body, let 

u be the distance by which a point, of which the coordinates 
are x, y, and z, is moved from its initial position, 

A the increment of a function for another point of the body 
which is near the former point, 

p the distance of the second point from the former point ; 

the notation of (42 ]2 ) gives 

p x = Ax=pCOS^., 

Ap x —p x D x u x -\-p y D y ii x -\-p z D z u x . 

Hence, if 

p ^p -\- Ap, 

J 

P 

a is the linear expansion of the body in the direction of p ; and 
its value is given by the equation 



( i+.).=®r=*® 



— 118 — 



— s ( pc "*" Apc Y 

= ^[(1 + D x ii x )cos p x -\- D y u x cos p -\- B z u x cos P ^ . 

J7j ^««, ^e reciprocal of 1 -J- e is laid off from the origin upon a line 
drawn parallel to p, its extremity ivill be upon the ellipsoid, of which the 
equation is 

1 = ^[(1 + D x u x )-k-\- D y u x j -\- B z u x zJ. 

221. The expansions or contractions which correspond to the 
axes of this ellipsoid may be called the principal expansions and con- 
tractions, and one of these is a maximum, another is a minimum, and 
the third is a maximum in some directions and a minimum in others. 

If the ellipsoid is referred to its axes, the expression for the 
expansion is, if e x , e y , and e z are the values of £ for the axes, 



(l + £ ) 2 = ^[(l+^)cos?] 2 . 



so that for these directions the values of u x , u y , u z , are such that 
(1 -f- D x ii x )D y u x -f- D x u y {l -J- D y u y ) -\- D x ii z D y u z = 0. 

(i + e x y = (i + D xUx y + {D xUy y + {D xUz y. 



222. The notation 



P 
<P=P> 



gives 



coscp = ^"Jcos^cos^ ) ; 
sin 2 (p = 1 — eos 2 (p 

= 2 X cos 2 p .2 x cos 2 x — Jj^Jcos^cos^ )| 

= 2 X ( cos^ cos z — cos z COS y ) 



i'\2 



— 119 — 

(1 -|~e) 2 sin 2 <jp = —,^ x (pcoSyCOsP — ji/cos^cos^ j 
= -2 2 * \p z cos J — p y cos z ) 
= Z x [(coslD x -f- co^D y -\- cos p z D z )(ii z cos J — z^cos J)J 

= ^|^:c( cos a^:J( M * cos w — «2,cosf ) I , 

in which the derivatives are only applicable to u x , u y , and u z . 
Hence, if the reciprocal of the square root of (1 -\- e) sin cp is laid off from 
the origin, upon a line draivn parallel to p, its extremity is upon the surface 
of the fourth degree, of which the equation is 

1 = Z x [Z x (xD x )(yu z — zu y )Y. 

223. When the axes are those of the ellipsoid of § 221, and 
the disturbance is such that for each axis the equations (118 19 ) and 
( USag) become 

£> y u x = 0, 

(1-f- e) 2 sin 2 g>— S x [cosjcos-? {D z u z — D y ii y )\ 
= ^|cos*cosJ(e JB --e y )J. 

224. To determine the rotative effects of the disturbance 
about the axes, let 



P 



and 



(f g = the projection of the angle (p upon a plane perpendicular 

to the direction of q. 
Hence 



<P* = Wx — V 



XI 



— 120 



, p, cos? 

1 Pv cos* 



. / i s »' D r i« 2 cos?-|-X)„?«,cos^ + (1 -4- D^i„) cos 
tan((p x + y x )=%- 



i>„ A^cosg + (1 + -A, u,J) cos J 4- D z u u qo^ ' 
225. If the axis of x is perpendicular toj», the equations are 
P — ip—P 

v — 2 F z 



Wx=W 



tanfo) 4- w ) — D » u * + (1 + AQtan^ 

226. If the axes and conditions are those of § 223, the equa- 
tion (120 3 ) becomes 

tan (9, -f ifj x ) = i^j'% tan^i. 

227. The whole expansion or contraction of the body at any 
time, is derived from the consideration that, by the definition of e in 
§ 220, any very minute portion of the body which is originally a 
sphere, becomes, in the disturbed state, an ellipsoid similar to that 
of §221. If, then, 

6 = the expansion of the body ; 

the sphere of which the radius is i, becomes an ellipsoid, of which 
the axes are i(l -|- e x ), i(l -j- e y ), i{l -\- t 2 ), and, therefore, its vol- 
ume becomes 

and 

i + a-(i + 4)(i + «,)(i+«.). 

228. When the disturbance is so small that the squares of the 
expansions may be neglected, which is the ordinary case of elas- 



— 121 — 

ticity, the equation (119 6 ) becomes 

e = S x cos 2 x B x u x -\- cos{,'cosf {D y u z -j- D x ii v ) I 
= S x (cos p x Z> x ) Z x (cos p x ii x y 

229. J7? //«?;?, #ta reciprocal of the square root of the linear expansion 
in any direction is laid off from the origin upon that direction, as the radius 
vector of a surface, the resulting surface is a surface of the second degree, of 
which the equation is 

1 = 2 x \x?D x u x -\-yz(D y u z -{- D z u y )~], 
or l=Z x (xD x )Z x (xu x ). 

230. If the axes are those of the principal expansions and 
contractions, the formula for expansion becomes 

6==^ (cos 2 J fcs); 

and the equations of § 221 become 

D x u y -{-D y u x =0, 



x "-"x • 



231. If the principal expansions and contractions are all of 
the same name, that is, if all are expansions, or if all are contrac- 
tions, the surface of § 229 is an ellipsoid. But, in other cases, in 
which, neither of the principal expansions is zero, the surface is the 
combination of two hyperboloids, of which one is one-parted, and 
the other is bi-parted. Both these hyperboloids have the same 
axes and the same asymptotic conical surface ; and the asymptotic 
conical surface, corresponding to the directions, in which there is 
neither expansion nor contraction, divides the directions in which 
the solid is expanded from those in which it is contracted. 

16 



— 122 — 

If one of the principal expansions is zero, the surface is reduced 
to a cylinder ; and if two of the principal expansions are zero, the 
surface is reduced to two parallel planes. 

232. In the present case, the formula of § 227, for the expan- 
sion of the solid, is reduced to 

ifl = e x -|- e y -f- e z = 2 x s x . 

233. The formula (120 9 ) for the rotation about the axis of x 
becomes, 

tan (w x 4- w) ■== tan w -I -^-f— 

= (1 + A«* — Dy u y) tan '/' — Attytan 8 l l> + D y u z , 
(p x = \{D y u z — Djt y ) -j- l(I) y u z -\-D z u y )cos 2y-\- \{p z u z — D y iiy) sin 2 y 
= IT x -\-t x cos 2 (y — tj x ) • 

in which 

JJ X = \ (D y u z — D z u y ) , 

r a cos2^ a = \{D y u z -j- ZU' y )> 
^ sin 2 ^ = -| (D z u z — D y u y ) . 

234. The maximum rotation about x corresponds, then, to 

W = q h 
and is y x = H x -\-r x i 

and the minimum rotation corresponds to 

is 

9* = H x — % x ', 

and Il x is the mean rotation. When the maximum and minimum 
rotations have opposite signs, there are two intermediate rotations 



— 123 — 
which vanish, corresponding to 

cos 2(y— «?«)= — — E - 

235. There are similar formulas for rotations ahove the axes 
of [I and g, and the combinations of the mean rotations give a great- 
est mean rotation, represented by 

the direction of which is determined by the equations represented 

by 

n tt x 

cos r =~. 

x n 

236. If the axes are those of § 230, the equations of § 233 
become 

II X = D y u z = — D z u y , 
y> x = IT x -\- I (D z u z — D y u y ) sin 2 x\> . 

237. When the disturbance is such that, for each of the prin- 
cipal axes, there is the equation 

D y ii z =0, 

the equations of the preceding section become 

H x = II=0, 

% = 1 ( A«* — DyUy) sin 2 y ; 

so that, in this case, there is compression tvithout any mean rotation. 

238. When the disturbance is such that for each of the princi- 
pal axes 

D r u x = : 



— 124 — 

the equations for compression and rotation becomes 

(p x = n x = D y u z , 

so that, in this case, there is rotation ivithout compression. 

All the preceding investigations upon the internal changes pro- 
duced by the disturbance of the form of a body are taken from 
Cauchy. 

239. The elastic force which is developed by any small dis- 
turbance of the internal condition of a body is proportional to the 
amount of disturbance, and has, therefore, the same general form 
with that of the disturbance itself. But the special discussion of the 
relative values of the coefficients involves the consideration of the 
laws of equilibrium, and must be reserved to a subsequent chapter. 



V. 

MODIFYING FORCES. 

240. Among the forces of nature, those which produce the 
equations of condition deserve peculiar consideration. Being 
merely conditional, they do not augment or decrease the power 
of a system, but merely modify its direction and distribution. They 
may, therefore, be called modifying forces ; and may be divided into 
two classes of stationary and moving. 

241. Stationary modifying forces are perpendicular to fixed sur- 
faces or lines, and constitute the action by which certain material 
points of a system are restrained to move upon those surfaces or 
lines. A force of this nature, being perpendicular to the motion of 



— 125 — 

its point of application, does not increase or diminish the total 
power of the system, but modifies its elements of direction. 
Thus the equation of condition, 

Z=0, 

between the coordinates of a point, involves the idea of a force, 
acting in the direction iVof a normal to the surface represented by 
this equation. When it is combined with its multiplier, it is equiva- 
lent, by (27i 6 ), (27 5 ), and (54 31 ), to a modifying force, of which the 
magnitude is 



o 



242. This force may be decomposed into three forces, which 
are parallel to three rectangular axes, either of which is represented 

by 

Xv/(nx)cosf, 

while the point of application moves through the elementary arc 
ds, its advance in the direction of the axis of x is 

ds cost. 

The amount of power added to the system, by the component 
force in the direction of the axis of x, is 

X ds y/ ( □ L) cos f cos* , 

and there is a consequent increase or diminution of force in this 
direction. But the mutual perpendicularity of iV^and s is expressed 
by the equation 

^(cosfcos*) = 0. 

The whole augmentation of power arising from the three com- 
ponents is, therefore, 

^^^(n^-^Cco-sfcos*) = 0, 



— 126 — 

which agrees with the fundamental conception of a stationary 
modifying force, and illustrates its mode of action. 

243. Moving modifying forces are perpendicular to moving sur- 
faces, which surfaces are themselves portions of the moving system, 
and the points of application are restrained to move upon these 
surfaces. In this case, the motion of each point of application may 
be decomposed into two parts, of which one part is perpendicular, 
and the other is parallel to the moving surfaces. The modifying 
force has the same relation to the motion which is perpendicular to 
it, which has been already discussed in reference to the stationary 
surface ; put by its relation to the other component of the motion, 
it communicates power to the point of application, or the reverse. 
But the power which is thus communicated to the point is 
abstracted from the surface, and through it from the other por- 
tions of the system ; and, therefore, the whole amount of power of 
the system is neither increased or decreased. Although for the pur- 
poses of theoretical speculation, it is convenient to regard the sur- 
face and the point of application as parts of one system, it is often 
the case in the useful arts that this transfer of power is of the 
highest practical importance, and is the basis of the theory of the 
turbine wheel. 

In a rigid system of bodies, these forces constitute the bonds of 
union. 



127 — 



CHAPTER VI. 

EQUILIBRIUM OF TRANSLATION. 

244. The conditions to which any combination of forces must be sub- 
ject, in order they may not tend to produce translation in the system of 
material points to which they are applied, are readily investigated. It 
follows immediately from §§18 and 20, and with the notation of 
those sections, that the algebraic condition that the system has no 
tendency to move in the direction of p is 

2[m 1 F 1 coaf = 0. 

But each term 

m 1 F 1 cosf , 

is the projection of the force m x F x upon the direction of p, and, 
therefore, if the algebraic sum of the projections of all the forces upon any 
direction vanishes, there is no tendency to translation in that direction. 

245. It also follows from the combination of translations, 
given in § 23, that if there is no tendency to translation in two different 
directions, ivhich are not parallel, there is no tendency to translation in the 
plane of these tivo directions ; and if there is no tendency to translation in 
three directions, ivhich are not in the same place, there is no tendency to 
translation in any direction. 

By means of rectangular axes the algebraic conditions, which 
are necessary and sufficient to produce equilibrium in respect to 
translation, are combined in the formula 

^ x [^ f 1 (m 1 F 1 cos})Y=0. 

This formula is independent of the situation of the points of the 



— 128 — 

system, except so far as the elements of position are implicitly con- 
tained in the expressions of the forces and their directions ; it would 
remain unchanged, therefore, if all the points were condensed into 
one, without any variation of the magnitude and direction of the 
forces. The conditions of equilibrium are, then, the same as if all 
the forces were applied at a single point. 

246. If one of the points of the system were subject to the 
condition of being confined to a fixed surface or line, the conditions 
of equilibrium of translation would simply be reduced to the condi- 
tion that the resultant of all the other forces would be perpendicular to this 
surface or line, and the modifying force by which the point was restrained 
would be equal and opposite to this resultant. 

If a point of the system was absolutely fixed, or if three differ- 
ent points were restrained to move upon three fixed surfaces, there 
would, in general, be no possibility of translation, but the resultant of all the 
forces applied to the system ivould be equal and opposite to that of the modi- 
fying forces by which the points were confined. 

247. The theory of the equilibrium of a point is wholly 
included in that of its translation. But since every system is a 
mere combination of points, the complete theory of equilibrium can 
easily be evolved from that of translation. This mode, however, of 
arriving at the conditions of equilibrium is neither luminous nor 
instructive. 

248. The conditions of the equilibrium of translation of a sys- 
tem, which is free from the action of all stationary modifying forces, 
may assume the form, that each force is equal and opposite to the result- 
ant of all the other forces. 

If, then, there are only two forces, they must be equal and oppo- 
site ; and if there are three forces, they must all lie in the same 
plane, and be represented by the sides of a triangle formed by three 
lines which have the same directions with the forces ; so that each 



— 129 — 

force must be proportional to the sine of the angle included between the other 
two forces. Whatever are the forces, if we were to start from a 
point, and proceed in the direction of either of the forces, through a 
distance proportional to the intensity of that force, and proceed 
again, in the same way, from the point at which we arrived in the 
direction of another force ; and so on, proceeding successively from 
each new station in the direction of the next force, through a 
distance proportional to that force, the course would finally termi- 
nate at the original point of its commencement. 



=♦= 



CHAPTER VII. 

EQUILIBRIUM OF ROTATION. 

249. The conditions to which a system of forces must be subject, in 
order that it mat/ not tend to produce rotation about a point or an axis, are 
directly deduced from §§ 84 and 88, and are simply, that the resultant 
moment of all the forces, with reference to the point or the projection of this 
resultant moment upon the axis, must vanish. 

250. When there is an equilibrium of rotation about a point, 
the resultant of the forces may not vanish, in which case there is 
not an equilibrium of translation. About any other point, there- 
fore, which is not situated in the line drawn parallel to the resultant 
through this point, there is not, by § 100, an equilibrium of rota- 
tion ; although there is an equilibrium of rotation about every point of that 
line. In order, then, that there may be an equilibrium of rotation about all 

17 



— 130 — 

points of space, or even about three points not in the same straight line, there 
must be an equilibrium of translation as tvell as of rotation. 

251. In the same way, it appears, that if there is an equilibrium of 
rotation about parallel axes lying in the same plane, there is an equilibrium 
of translation in the direction perpendicular to the plane ; and if there is 
equilibrium of rotation about parallel axes which are not in the same plane, 
there is an equilibrium of translation in every direction except that of the 
parallel axes. 

252. If there is a fixed point in a system, it is necessary and 
sufficient for the equilibrium of rotation that the resultant moment for this 
point should be nothing ; and, in this case, the resultant moment vanishes for 
every point of the straight line which is drawn through the fixed point par- 
allel to the resultant, and also for every axis ivhich is in the same plane with 
this straight line. 

253. If there are two fixed points in a system, it is necessary 
aud sufficient for the equilibrium of rotation that the moment of the forces 
should vanish for the line which joins the two points. 

254. If all the forces are parallel and equal, there is, by § 99, 
combined with § 250, a line parallel to the common direction of the 
forces for which the resultant moment vanishes. If the common 
direction of the forces is assumed for that of the axis of z, the 
moment of the force acting upon a particle dm, with reference to an 
axis drawn parallel to that of y at the distance a, from the plane of 

yz, is 

{x — a)Fdm, 

and the whole moment of the system is 

f (x — a)F=Ff (x — a). 

U m O m 

The condition therefore that the moment vanishes for this axis is 



/{% — a) = ; 
m 



— 131 — 

and the plane which is thus drawn at the distance a from the 
plane of yz, includes, by § 127, the centre of gravity. Hence, 
the axis, for which the resultant moment of the parallel, and equal forces 
acting upon a system vanishes, passes through the centre of gravity ; and if 
the system has an equilibrium of rotation, and if there is a fixed point in it, 
the centre of gravity must he in the straight line which is drawn through the 
fixed point in the common direction of the forces ; or, if there is a fixed axis, 
the centre of gravity must lie in the plane which includes this axis and the 
direction of the forces. It is also apparent that, if the centre of gravity is 
advanced beyond the fixed point or axis in the direction of the forces, the 
equilibrium is stable ; but if the centre of gravity is not so far advanced as 
the fixed point or axis, the equilibrium is unstable. 

The ordinary case of gravitation at the surface of the earth, in 
which its variation in intensity and deviation from parallelism is 
insensible for the small system of bodies discussed in the usual 
investigations of mechanics, is the familiar type of this species of 
force. 

255. In the motions of translation and rotation there is no 
motion of the parts of the system among themselves. There is no 
change, therefore, in the mutual distance of the origin and point of 
application of each of the forces which arise from the action of the 
parts of the system upon each other. The origin, regarded as a 
point of application of the same force, acting in the opposite direc- 
tion, moves just as far in the direction of the force as the actual 
point of application ; so that such a force acts precisely as a moving, 
modifying force, and has no tendency to affect the equilibrium of 
translation or rotation. All the forces, therefore, between the different 
parts of the system may be neglected in determining the conditions of the equi- 
librium of translation or rotation. 

This mutual relation of the origin and point of application of 
the force, by which either may be regarded, at pleasure, as being 



— 132 — 

the origin or the point of application, by a simple reversal of the 
direction of the force without any change of its intensity, is com- 
monly expressed by the proposition that action and reaction are equal. 



CHAPTER VIII. 
EQUILIBRIUM OF EQUAL AND PARALLEL FORCES. 



I. 

MAXIMA AND MINIMA OF THE POTENTIAL. 

256. In orcler to give precision to the modes of expression, 
and have the benefit of well-known terms and forms of speech, the 
force considered in this chapter, is assumed to be the typical force 
of gravitation at the surface of the earth, acting within a space small 
enough to admit of the neglect of its variation of intensity and devicdion 

from parcdlelism. 

The level surfaces of this force are horizontal planes, and the potential 
decreases uniformly with the increase of height above the earth's surface. 

257. Let the three rectangular axes be so assumed that the 
plane of xz is horizontal, the axis of y, the upward vertical, that of 
x, the northern horizontal line, and that of z, the western horizontal 
line. If, then, 

g is the intensity of the force of gravity, 

G the distance of the centre of gravity from the origin, and 



— 133 — 

I2 the value which the potential would assume, if all the points 
were in the plane of xs ; 

the actual value of the potential is, by the property of the centre of 
gravity, 

£l = S2 -f y = S2 -f fr—G,+ G,) 
= I2 — / G S = S2 — niG y . 

U m 

Hence the potential is a maximum, when the height of the centre of 
gravity is a minimum, and such a position of the system corresponds, by § 62, 
to that of stable equilibrium ; but the potential is a minimum, when the height 
of the centre of gravity is a maximum, and such a position corresponds to 
that of unstable equilibrium. 

258. Since the direction of gravity is the same for all the 
points of the system, there cannot be an equilibrium of translation, unless 
there are stationary modifying forces, the resultant of ivhich must be exactly 
equal to the whole weight of the system, and have a vertical, upward direc- 
tion. 

2-59. The resultant moment of all the forces of gravity van- 
ishes for the centre of gravity ; and, therefore, the resultant moment of 
all the stationary modifying forces must vanish for the same point. 

260. If there is but one modifying force in the system, it must 
be vertically directed upnvarcls, have an intensity equal to the ivhole iveight of 
the system, and its line of action must pass through the centre of gravity. 

261. If there are but two stationary modifying forces, they 
must lie in a common plane, ivhich is vertical, and includes the centre of 
gravity, their resultant must have an upward direction, and be equal to the 
weight of the system, and they must be reciprocally proportional to the dis- 
tances of their directions from the centime of gravity. This last condition is 
involved in the necessity that the resultant moment must vanish 
for the centre of gravity. 



— 134 — 

262. If the intensity of the force of gravity were to be 
increased or diminished, the conditions of the position of equilib- 
rium would not be changed, but intensity of the modifying forces 
would be proportionally increased or diminished. Even if the force 
of gravity were to be made negative, that is, if the direction of its 
action were to be reversed, the conditions of the position of equilib- 
rium would still remain unchanged, provided that the modifying 
forces were of such a nature that the direction of their action would 
also be reversed ; but, in this case, the position of stable equilibrium 
becomes that of unstable equilibrium and the opposite. This rever- 
sal of the direction of gravity is relatively accomplished by the 
rotation of the whole system about a horizontal axis. 



II. 

THE FUNICULAR AND THE CATENARY. 

263. When the points of application of a system of forces are 
united by a single continuous chord which is destitute of mass, the 
polygon, which is formed in the situation of equilibrium, is called a 
funicular. The general conditions of such a system involve a mere 
repetition of the principles of equilibrium ; and the present discus- 
sion is limited to the case, in which the points of application are 
masses acted upon by gravity. 

264. When there is but one fixed point to the system which 
may, without any essential loss of generality, be assumed to be 
either extremity of the chord, in every position of cquilibriumjlie chord 
must be vertical. 

But if the idea of the incompressible rod is supposed to be 
included in that of the inextensible chord, each portion of the chord 
included between two successive masses may be assumed to have a 



— 135 — 

vertical direction, either upwards or downwards ; so that, if 

n is the number of masses, 
2 " is the number of positions of equilibrium, 

all of these positions, except that one in which every portion of the 
cord is directed downwards, involves an element of instability, and 
must, therefore, be regarded as absolutely unstable. The tension of each 
portion of the chord is, in every case, equal to that of all the tveight which it 
has to sustain ; that is, to the sum of all the subsequent masses ivhich lie 
upon the portion of the chord not attached to the point of suspension. 

265. When there are two fixed points, the whole included 
chord must hang in the same vertical plane with these two points. 
The tensions of the various portions of the chord represent modifying 
forces ; and the surfaces at which these forces act are those of 
spheres, all the centres of which are movable, except those of the 
two fixed points. In the position of equilibrium, however, all the 
centres become stationary, and the conditions of equilibrium of each 
mass or portion of the chord admit of independent discussion. 

The forces which act upon each mass are gravity and the ten- 
sions of the two portions of chord upon each side. The horizontal 
projections of these two tensions must, therefore, be equal and oppo- 
site in order to balance each other ; so that the horizontal projection of 
the tension of the chord is invariable throughout Us whole length, and equal to 
the horizontal projection of the sustaining force of each of the fixed points. 

The algebraic sum of the upward vertical projections of the tensions at 
the two extremities of any portion of the chord must be equal to the ivcight of 
all the intermediate masses in order to support them against the force 
of gravity. 

266. These two conditions are necessary and sufficient to 
produce an equilibrium of translation in any portion of the chord, 
and, therefore, of the whole chord. The condition of the equilib- 



— 136 — 

rium of rotation of each portion of the chord, although included in 
the preceding conditions, is an interesting and useful modification of 
them. 

With reference to the centre of gravity of the masses of each 
portion of the chord, the moment of the gravity of the masses is 
zero, and. therefore the moment of the tensions applied at the 
extremities must also vanish. But the directions of these tensions 
are not parallel, and therefore their lines of tension produced must 
meet at a point, at which both the tensions may be regarded as 
applied without affecting their tendency to produce rotation. At 
this new point of application they may be combined into a result- 
ant, which is vertical, because the horizontal projections of the ten- 
sions are equal and opposite. This resultant has the same tendency 
to produce rotation with the tensions themselves, and therefore it 
must pass through the point for which this tendency vanishes, 
that is, through the centre of gravity of the masses. The point of 
meeting, therefore, pf the lines of extreme tension of any portion of a chord 
is in the same vertical ivilh the centre of gravity of the intermediate masses. 

267. If the two extremities of any portion of the chord are 
in the same horizontal line, the equal horizontal projections of the 
extreme tensions are exactly opposed, and therefore the moments 
of the vertical projections of these tensions must be equal with 
reference to the centre of gravity. The vertical projections of the 
extreme tensions of any 'portion of the chord, of which the extremities are 
upon the same horizontal line, are, then, reciprocally proportional to their 
distances from the vertical drawn through the centre of gravity of the inter- 
mediate masses. 

268. Since the horizontal projection of the tension of the 
chord is the same throughout its whole extent, no portion of the 
chord can become vertical. If any portion of the chord is hori- 
zontal, the vertical projection of its tension vanishes, and, therefore, 



— 137 — 

the vertical projection of the chord at any other point is equal to 
the sum of the weights of all the masses intermediate between this 
point and the horizontal portion. If then 

T is the tension of the chord at any point, 

and if the axis of x is horizontal, and that of y vertical, directed 
upwards, so that 

T x is the horizontal projection of T, and 
T y its vertical projection ; and if 
s is the arc of the chord at any point, and 
m the sum of all the masses included between the point and 
the horizontal portion of the chord ; 

the following equations express the preceding conditions : 

Tcos* = T y = in, 

, s __ m 
tan x — ypr . 

The inclination of the chord to the horizon, therefore, increases 
as the distance recedes from the horizontal portion. 

If the chord has actually no horizontal portion, the preceding 
equations are still applicable by assuming for m, such a value as 
would be required to correspond to the vertical tension of any given 
portion of the chord. 

269. If, in proceeding from the horizontal portion in either 
direction, the chord is everywhere ascending or descending, its hori- 
zontal direction must also be away from the extremity of the hori- 
zontal portion to which it is attached so as to form a portion of a 
convex polygon, which cannot be intersected more than once by 
any vertical line. Such a position of the chord corresponds to that 

18 



— 138 — 

of the perfectly stable state, or to that of the most unstable state ; 
and each state is always possible. 

If, in proceeding from the horizontal portion, the direction of 
motion changes from ascent to descent, or the reverse, the horizon- 
tal direction must be reversed at the same time, and so that the 
subsequent portion of the chord will form an arc of a polygon which 
will include the preceding portion within its concavity, and the con- 
cavities of both portions will be turned the same way. 

270. The difference of equation (137i 8 ) applied to two differ- 
ent portions of the chord gives the following equation between the 
intermediate masses, the horizontal tension, and the directions of 
tension at the two points, 

m' — m sin ( s x ' — s x ) 

T x cos %' cos % 

271. If the masses are infinite in number, and arranged in 
unbroken continuity so as to form the chord itself, the curve is 
called the catenary. In this case, if 

k is the weight of an unit of length of the chord, the mass 

of an element is 
dm = 7cds ; and if 
o = the radius of curvature, 

the equation (138 13 ), applied to the extremities of the element, 
gives, for the equation of the catenary, 



<? = 


T 

zzD«s — -f sec 2 x 


If 


T 

A — -i 


this equation becomes 





D* x s = q = J.sec 2 *. 



— 139 — 

272. If the chord is of uniform thickness and density throughout its 
length, Jc and A are constant, and the integral of (138 31 ) is 



s = A tan 



XI 



to which no constant is added, because the arc is supposed to be 
measured from the point at which it is horizontal. 

273. The curve of the uniform chord is easily referred to rec- 
tangular coordinates, for the equations 

Dsjj — D.^ssin s x = Asm s x sec 2 *, 
D* x x = D x scos x — .Asec * ; 

give, by integration, and determining the constants, so that the ori- 
gin may be at the point of horizontality, 

y = A(sec s x — l), 

a; = ^logtan-|-(i7r— •) . 

These equations give, by elimination and the use of the nota- 
tion of potential functions, 

Sin^ = tan x s =:^, 



5 = 00^-1 = ^ + 1)- 1, 



274. The vertical tension of the uniform chord is 
T y = sk = ±T x = T x tm x = T x Sm^; 
and the whole tension is 

T= T x secZ= T x Cos^ = T X (L + l) = T x) J±. 



— 140 — 

275. If the chord were required to he of such a variable thickness as 
to assume a given form of curve, the law of this variable thickness is 
given by the equation 

T 

'*• — 2! • 

pcos 2 ^. 

The vertical tension is 

■L v == SK == il ) 

and the whole tension is 

T=T x sec x . 

276. If the thickness of the chord ivere required to he proportional to 
its tension, so that 

T 

the following equations are successively obtained by easy transfor- 
mations 

Bfs = B sec x , 

D° x z = B, z = B(Q, 

Sin -^ = tan * = tan ^, 

| = log sec J = log Cos ^, 
q = Bsec -„ = BCos-jj= c y , 
T= T x sec x =T x sec^ = T x Cos^ = T x c\ 

277. If the thickness he such as to give an uniform horizontal distri- 
bution of the weight, that is, such a distribution that the weight of each 
portion of the chord is proportional to its horizontal projection, the 
equations are 



— 141 — 

D*s = q = Cscc 3 x, 
x = Ctan *, 
y = 1 67(sec 2 ^ — 1) = \ Cton\ = ^ ; 

and the curve is a parabola, of which the transverse axis is vertical. 
278. If the chord were compressible and extensible, it would 
be compelled to assume that thickness, in which it would have the 
requisite tension ; and the form of the curve would, with this condi- 
tion, be the same as if it were incompressible and inextensible. 
Thus, if F denotes the function which expresses the given law of 
the relation of the thickness to the tension, so that 

the form of the curve is given by the equations 

l 



D iS == q = 

*>& = 

D°x = 



cos 2 £F {sec 2 J) ' 
sin' 



cos 2 ^(sec 2 £)' 
1 



cos^i^sec 2 ^) 



279. If the chord or any portion of it is confined to a given 
surface, the resultant of gravity and the tension of the chord on 
each point must be normal to the surface, and is balanced by the 
modifying force by which the point is fixed to the surface. 

If, then, the tangent plane to the surface is, at each point, 
assumed as the plane of x ' y' ; if the axis of x' is horizontal, and that 
of y directed upwards, and if 

(/ is the radius of curvature, at this point, of the projection of 
■ the chord upon this plane ; 

the curve and tension may be determined by means of the equa- 



tions 



142 



T T„, 



s ■ 


k&m s y 


rCOS^/ 


&cos 2 £.cos^. 


> 






T . 


r = QsecP, 






&sin*, 


COS*, COS* 


J 


D S T: 


= #COS 


2,/COS^/ = 


= #COS* = 


JcD s y, 


T-. 


J y 









280. The pressure upon the surface is determined by the con- 
sideration that it must exactly balance the tendency of each point 
of the chord to move in the direction of the normal to the surface. 
But the tendency of the tension to move any point of the chord in 
any direction, as that of p, is 

D s T p = I) s (Tcos;) 

= cos; D S T— Tsm; D s s p . 

In the case of the direction N of the normal to the surface, this 
expression becomes, because s is perpendicular to N, 



T 

Us J- N z= J- U s N = -j, 



Tcos ? 



N . 



in which 



q" is the radius of curvature of the projection of the chord 
upon the common plane of the normal to the surface, and 
the tangent the chord. 

Hence the pressure sustained by the surface in the direction of 
the normal is 

Q ' 

281. If the chord is destitute of weight upon any portion of the sur- 



— 143 — 

face, q' becomes infinite, and the carve is that of the shortest line which can 
be drawn upon the surface. 

The tension, in this case, is constant, and the pressure upon the 

surface becomes 

T 
R = -. 
Q 

282. In the case of a cylinder, of which the axis is vertical, the 
equations become 

y u ? 

T 

T 

so that the curve is the same ivhen it is developed ivith the cylinder into a 
plane, which it assumes ivhen it hangs freely. 

283. In the case of a surface of revolution about a vertical axis and 
a chord of uniform thichiess, the equations become 

T=lc{y +!/»), 
r j y + y<> . 



sin „, cos 



y > 



in which the angle which y makes with y' is determined by the 
meridian curve of the given surface, the plane of xz passes through 
the lowest point of the curve, and y is the length of the chord 
which is equal in weight to the tension at the lowest point. 

284. A special solution of the preceding problem is given by 
the equations 

y=0, \, = \n, 



o = 



cos 



The curve is the circumference of the circle formed by the intersection 
of a horizontal plane tvith the surface of revolution. The tension of the 



— 144 — 

chord is the iv eight of a length of the same chord which is equal to the dis- 
tance of the plane of the curve from the vertex of the cone drawn, through the 
curve, tangent to the surface. 

285. If 

tp is the angle which the projection of y' upon the plane of 

xz makes with the axis of x, and if 
dui' is the elementary angle which two successive positions 

of y' make with each other, 

this elementary angle and the radius of curvature are given by the 
equations 

dy'= $ml,dy, 
1 = — D t \, + Dtf = smpAy — D* 

= sin*, D s y — cos y ,D yfy , = sin*, D s y — Z^sin*,. 

If, moreover, 

u' is the length of the tangent drawn to the meridian 

curve at any point of the chord, and 
u the projection of u' upon the axis ofy, 

the following equations are obtained, 

sin s y , = u' . sin y y , D s xjj -= u' D s \p' , 

_ an,, _ D ^ m ^ _ s i n 5 /COS |^_ _ 2y gsin*,) ; 

which substituted in (143 19 ) gives, by dividing by sin s y , cos v yf , and 
transposing, 

286. In the case of the right cone, ivith the circular base, the sum of 



— 145 — 

y' and u' is constant ; if, then, 

a = u + y, 

a = u -\-y = a'cos^, ; 

the curve is determined by the equation 

Z> , lo^ sin', = -i- : — > = — -4- -7 t > 

= — i^log.sm.^. 

The integral of this equation is 



snu, = 



a'Vo 



u 



' («' — a' - fa\) " («' + tf) a - («' + ^ - 2 O 2 



(«' + yC) 2 -(a'-^) 



'\2J 



in which the constant is determined, so that «' may be equal to d 
when the chord is perpendicular to u . 

The chord is also perpendicular to u', when 

and also when 

u = h («' + *» ± i V [(« + I/oY + 4 «>o] • 

When w' is contained between a' and ^? the expression for the 
sine of the angle which the chord makes with u' is less than unity, 
so that the angle is real. This angle is also real when ii surpasses 
the greater of the roots of (145 2 i), or when it is algebraically inferior 
to the smaller of those roots ; but the angle is not real when ii is 
included between these roots, but is exterior to the preceding limits 
ii and y' . The curve of the catenary upon the vertical right cone consists, 
therefore, of three distinct portions, of ivhich one is finite, and included 
between two intermediate points, at ivhich the curve is perpendicular to the side 

19 



— 146 — 

of the cone ; ivliile the other huo portions, commencing respectively at the tivo 
points, which are the highest and lowest of those at which the curve is perpen- 
dicular to the side of the cone, extend to an infinite distance. These portions 
have tivo of the sides of the cone for their asymptotes, because the angle 
which s makes with u' vanishes, when itf is infinite. 

287. The finite portion of the catenary upon the vertical right cone 
may be investigated by adopting the notation 

siny = V" , — ■, 
sin 2 2= cos 2/3, 

sin i _ a' — rf 

n — i — 7 i 7 1 
cosp a +3fo 

sin$ = sun sin <p ; 

and that of elliptic integrals, of which the third form may be repre- 
sented by 

sec# 






a 1 -f- n sin * (f 
v 

These equations give 

u'= k (a' -\-y' )(l — sin i sec /? sing)) 
= %(a' + y' )(l — sec(i$m6), 

cos 2 ^ — sin 2 i sin 2 /3 



sm 



"' cos 2 |3 — sin 2 cos 2 # — sin 2 ^' 



cos \J (cos 2 (9 — 2 sin 2 (3) _ sin i cos # cos qp 
COS„, = ■ cos'fl — sin 3 |J ~ cos 2 <9 — sin 2 ^' 

s sin 2 (3 

tan , / — -: — -. — , 

sin 1 cos a cos <p 

Z>, m' = — ^ {a! -{- ^0) sin «'sec cos y , 

n s — _ ^W _ i Q' + ?A')Qs 2 fl — sin'fl 
cos*/ - cos ^ cos 

= \ ( a ' 4" #0 ) ( sec /? cos A — tan /J sin ft sec 6), 



w 



— 147 — 

s = I ((i-\-!/o) (sec/? % — tan/? sin/? 9? 4 g>) ; 

tan;%Z)(i,?/ tan (S sin (3 tan (3 sin (3 



M 



(1 — secp'sin^cosfl (1 — n sine/) cos 



tan/3 sin |S sec , sin* tan 2 /? sin gp sec 



lanpsinpsucv , suit 
1 — n - sin 2 9 "^ 1 



• ?« 2 sin 2 <jp 



f = tan/? sin/? 0, (- rc 2 , 9 ) + tan'" 1 ^ ; 
for it is found, by differentiation, that 

V tang) ? cos<9 

_ sin i sin rp (cos 2 — sin 2 i cos 2 qp) 
(cos 2 d -\- sin 2 i cos 2 cp) cos 

sin i tan 2 f> sin q> sec d 
1 — ?« 2 sin 2 cp 

288. The preceding value of the angle \\>' admits of geometri- 
cal expression by means of the arc of the spherical ellipse in the 
form given by Booth. 

A spherical ellipse is the intersection of a cone of the second degree ivith 
a sphere of which the centre is the vertex of the cone. Let 

a and /? be the two principal semiangles of the cone, of which 

a is the greater, and 
10 the angular distance of any point of the arc of the ellipse 

from its centre ; 

and its equation is obviously 

4.2 1 cos2 a i sin 2 £ 

C0t 2 W = — — = -^ -\ j£. 

tan^w tan^a ' tan^p 

Adopt the notation 

o = the arc of the spherical ellipse, 



— 148 — 

i = the angle which the perpendicular to either of the cir- 
cular sections of the cone makes with the axis, which 
perpendicular is called the cyclic axis, 
e = the angle which the focal of the cone makes with the axis, 
rj = the angle of eccentricity of the elliptic base of the cone. 

If, then, through the centre (fig. 2) of the spherical ellipse, 
the axes A OA and B OB' are drawn, and B joined to the foci F 
and F', the sides and angle of the spherical triangle B OF, are 

BF=a, B0 = (1, OF=t, 
OBF=i], BFO = \n — i, 

which are connected by the equations 

cos a = cos (i cose = cot rj tan/, 
sin ft = sin a cos i = cot rj cot e , 

sine = sin a sin rj = tam'tan/?, 
costj = cos icose = cot a tan/?, 

sin i=8in rj cos (i = cot a tan e . 

Let C and C be the points at which the cyclic axes cut the 
surface of the sphere. Draw OF to any point of the ellipse, OF 
perpendicular to OF, CH perpendicular to CF, OH perpendicular 
to CH, F'K perpendicular to OH; take F'K equal to OC, and 
draw LM perpendicular to OA. If, then, 

6 = LM, 9 = LF'M, 
l = HOC, l'=OCF, 

the following equations are readily obtained, 

cos/= cos 0C= coU'tan£, 
tan£ = cos a' tan I' = cos 2 /tan I 

= cos 2 i cose tan (p = cos i cos rj tan <p , 



— 149 — 

sin 6 = sin i sin cp , 
sec 2u __ ]_ _j_ cos 2 /cos 2 jj tan 2 <p = sec 2 (p(cos 2 y -J- cos 2 2COS 2 ^sin 2 <p) 
= sec 2 c/>(l — sin 2 j;sin 2 g) -f- sin 2 ^sin 2 0) 
= sec 2 9)(cos 2 d — sin 2 ?] cos 2 i sin 2 y), 

9 9 9,, /i i 2 -. 9 \ 1 -I- cos 2 i'tan 2 qp 

cos 2 w = cos-«cos 2 "(l + cos Man ''cp) = - — = — ^- — ^ — r^—r- > 

2 _ cos 2 «(l -(- cos 2 ^tan 2 qp) _ cos 2 «cos 2 

1 -(- cos 2 // tan 2 cp 1 — sin 2 // sin 2 qp ' 

. o sin 2 «cos 2 cpsec 2 ^ 

snr&i = 



1 — sin 2 ?/ sin 2 qo ' 

r 



_ cos ?/ cos » cos 2 £ _ cos // cos t sin 2 « 



cos 2 qp sin 2 w(l — sin 2 '// sin 2 </) 

^ _ cos 2 « sin 2 //sin 2 (5 sin qr cosqo 

(1 — sin 2 // sin 2 <jp) 2 sincocos(u' 

y, 2 sin 2 (3 cos 2 // sin 2 // sin 2 1 sin 2 9 cos 2 " 

^ cos 2 0(l — sin 2 //sin 2 qp) 2 

? a cos-qp(l — sin //sin-qc) 



<S> 



sin 2 (3cos 2 // /cos 2 5sin 2 //sin 2 qpcos 2 -|- sin 2 ?/sin 2 5> 



cos 2 5(1 — sin 2 // sin qr) 2 \ cos 2 qpsec 2 £ 

sin 2 |3cos 2 // /cos 2 6 — sin 2 //cos 2 '<sin 2 gA 

cos 2 0(l — sin 2 //sin 2 93) 2 \ cos 2 qpsec 2 £ / 

sin 2 |3cos 2 ?/ 
cos 2 0(l — sin 2 j/sin 2 qp) 2 ' 



r. sin B cos i] sec 6 

V 1 — sin 'tj sin qo 



a = sin/9cosi7 < 3 s i ( — sin 2 ^, y) = ta "^ sin ' ffi.( — sin 2 »j, 9). 
289. In the particular case in which 



in, 



this equation is, by (148 17 ), reduced to 

a = tan^sin^°J s i (— n*,-q>), 



— 150 — 
which substituted in (147 5 ) gives, 

/ i , r— 11 tan 

1 ' tan <jp 

290. For the length of the arc of the chord which extends 
from its lowest to its highest point, this equation becomes 

2tf = 2a i; 

and if the magnitude of this angle is commensurate with the total 
developed angle of the cone, the chord returns into itself, after passing 
around the cone once, tivice, or several times, dependent upon the magnitude 
of the angle of the cone. 

291. To investigate the infinite portions of the chord, let 

Iq and l[ be the roots of the equation (145 21 ), 
and the equation gives 

7' f — ' ' 

h n — a l/o • 
Adopt also the notation of § 287 and 

(«' — ffo) _ a' + tfo _ (« + K) sec P 



smfjp 



smi(a'-\-f —2u') cos (a' + f u — 2 v!) l' J r l[ — 2v! 



suit) = sim sin 9, 

and the following reductions are obtained, by the substitution of 
cosecg/ for sin (3, 

u' = | (/q -j- l[) ( 1 — sec /5 cosec 9') , 

77 ,-lW I „M / tan ft sin ft Bec/?cos 3 qp\ 
% /S — 2 l« -h ^0 j ^~^7 un'qfcaatf) 

= |(a'4-#o)[tan/5sin/5sec0'-f- sec/5cos£'-}- sec/?Z^,(cos3'cot(p')] , 
s = 1 (a'-j-^o) (sec/5 ^g>'-|- tan/5 sin/5 S^g/-}- sec/5 cosd'cotg/), 



7-1 I SI 

D ¥ y =- 



— 151 — 

sin - § cos (j sin 2 g/ sec 6' — sin 2 ft sin g/ sec 0' 



1 — cos 2 /? sin 2 g/ 



1 — cos-jism-y ' ' ' 9 cosgi " 

cos0' 
cos g/ ' 



i// = tan /? sin /? <3>, (— cos 2 ft tp') — tan (i sin /if ^ c/ -f- tan [ ~ 1] 



292. The term of the preceding value of y>', which depends 
upon elliptic integrals of the third order, may be constructed by 
means of a spherical ellipse, of which the parameter is the reciprocal 
of that employed in the construction of the similar term of the finite 
portion of the chord. The parameter of the spherical ellipse of 
§ 287 being sin.], the reciprocal parameter is 

sin i ,, 

- — = cos i , 

sin tj ' ' 

and the length of the arc of the corresponding spherical ellipse for 
the amplitude (p r is 

a' = sin/J cosi^C— cos 2 /*, 9') = tan t ^ n/? Q J.(— cos 2 /?, 9'). 
This arc is reduced, in the case of 

to 

a' = tan sin % (— cos 2 /J, 9') . 

293. The finite portion is exactly circular tuhcn 

a' = / . 
In this case 

2 = 0, /$ = £ 7t = « , 

and the equations of the infinite portion become 

u' = d (1 — y/ 2 . cosec 9') , 
s = asJ2 cot 9/ 
i / /= v /2(i7T — 9/) — 2tan [ - 1 i[( v /2 — l)tan(|7t — }f//)]. 



— 152 — 

294. As y' diminishes from the value a , the finite portion 
becomes more and more eccentric, until when 

both the finite and the infinite portions degenerate into straight lines, which 
are the sides of the cone. 

295. When y' is negative, a! and y'^ cease to be the limits of the finite 
portion, and become the limits of the infinite portion, while I Q and l[ become 
the limits of the finite portion. But 1' and l[ are imaginary, if y is included 
bettveen the values 

/ =(-3 + 2v/2-K 

so that betiveen these limits the finite portion disappears, and the chord con- 
sists only of the tivo infinite portions ; and at the limits the finite portion is 
circular. 

To investigate the infinite portions betiveen the limits, in ivhich the finite 
portion disappears, let 

r . . . 

tan i' — sin i\J — 1 , 

sin 6" = sin i' sin y" ; 

and the following equations are obtained by simple transformations, 

sin (i cos/ = y/-|, 

u / =l-(a / -]-yo)(l — sec /3 sec g/') = -|-(a' — #o)(cos/3 — seer/), 

cos 2 £'=l-}- tan V cosy 

= secY(l — sinVsm 2 <j>") 

= sec*Vcos 2 £"; 

B ~ — i (V I W cos*''sec/?sinV tan/? \ 

■*>' s — if \ a -T l/o) \ cos * yii cos Q n ^ 2 . cos 6") ' 

,/ / , ,,/secB .„ „ sec/?© // t.an/?^ ,A 

» = *-(« +^o)(^cos^ tany -^«*r? — p-^ji 



— 153 — 

9 * 1 — cos-p i cos- y ' ' - ' r tantf 

cos »' cos 8 sec 0" , ./ ,, ,« v 

= . '. „ „ + cos? cos 8 seed 

1 — i sin - q> ' ' 

- Z^ tan<-«(cos/Jcosy*^ - /?^tant-ii^, 

i/;' = — cos/cos/3°3 i i ,( — I, y") -J- cos /cos/? 9^9" 

— tan [_ 1] ( cos 8 cos 9" -^-^ ) — tan [_ 1] -^. 

\ ' 7 tain/ tan0 

= — cos /cos/? ^i, ( — i, g>") 4" cos /cos/? 3^ 9" 



_ r-11 tanV-fcos/?cosqr/'tan 2 0" . 

tan i' tan0"(l — cos 8 cos g/') ' 

in which the elliptic integral of the third form admits of interpreta- 
tion by means of the arc of the spherical ellipse. 

296. When the negative of y' is equal to a the equations may 
be greatly simplified and reduced to the following forms, 



P-- 




In 

2 J 


2 = 


-\n, 








cos/? 


= 


o, 


COS?' 


=Vi, 








Dtf'tf 


— 




cos q!' 










V/(2- 


— sin 2 


<*")' 




sin i// 


= 


sin qp 


n 










V/2 




COS 2 (f" 




1- 


- 2 sin 2 i// = 


COS 


2 


/, 


„ '2. 

u 




a"- 




a' 2 


j 






cos 2 


v"~ 


cos 2 a// 





w/zj'cA 25 the polar equation of the equilateral hyperbola. In this case, there- 
fore, the curve of the chord upon the developed cone is an equilateral hyper- 
bola ; this case was recognized by Bobilliee in an imperfect investi- 
gation of the catenary upon the surface of the vertical cone of revo- 
lution. 

20 



— 154 — 

297. When the surface of revolution is an ellipsoid, of which the 
equation of the vertical section made by the plane yx is, 



y__i_£_ 



let a sphere be constructed upon the axis of revolution as a diame- 
ter, and let 

y be the angle from the vertical point of the sphere to a point 
of which y is the ordinate, so that 

^ = -4 cosy, x = B sirup, 
u = .4 (secy — cosy) = A sin (p tan cp, 
D^y = — As'm(p. 

These equations, substituted in (144 29 ), with proper regard to 
the different position of the origin of coordinates, give 

iyogsin*, = — ^sin^Zyogsin*, = —cotcp -4- C J™^_ M , 

• s _ jv _ iy 

y> sin qp (cos cp -\- M) \ sin 2 qp -|- iJ/ sinqp ' 

in which JVand Jf are arbitrary constants. 

298. The maximum and minimum of sin*, are determined by 
the roots of the equation 

cos 2 cp -f- Mcosy = 0. 

If these roots are y' and cp", the equation gives 

cosy' cosy'' -4-^ = 0, 

J^^ — 2 (cosy + cosy") = — 4 cos a (y' + y") cos -| (y' — y") 
= secy/ -J- secy". 

Of the two roots, therefore, one is obtuse, while the other is 



— 155 — 

acute ; if one is contained between i n and §7t, the other is impos- 
sible ; and when both are real, one is confined between \ it and f it , 
while the other is without these limits. The corresponding mini- 
mum and maximum values of sin*/ are 

N.N 
-, and 



tan qp' sin 2 qp' tan qj' sin 2 qp" ' 

Both these, independently of their signs, are minimum values, 
and when they are both absolutely greater than unity there is no 
catenary ; but if either is less than unity, there is a corresponding 
portion of the catenary. When both values are less than unity, the 
catenary consists of two separate portions, because there is between 
<p' and y" a value tp'" of tp which satisfies the equation 

cosg/" = — M, 

and the values of 

/ /// cos 2 m' — cos 2 q> sin 2 a! 

COSCP COSC/) = i -. ?- = -v, 

' ' cos q> cos 9 

/// // Sin qp n ' v rr r 

cos(p — cosy = 7-V— 2sm z (p cosy, 

arc positive. 

299. TVic especial case of 

gives 

Jf=0; 

sin£,= -^— ; 

* sin 2 qp 

and each of the minimum values of sin*, is 

2JST; 



— 15G — 

which, being less than unity, may be expressed by 

2iV=sin2«. 

This equation gives 

. s _ sin 2 a 

Sin,./ ~ t: . 

J SHI 2 (J) 

If, then, X is determined by the condition 

, cos 2 cp 
cos 2 /. = — »— , 

cos 2 « 

simple reductions give 

_ \j (cos 2 2 « — cos 2 2 r/i) cos 2 a sin 2 ?. 



UUBj/ — 


sin2qp 


sin 2 cp ' 




tan*/ = 


tan 2 a 
sin 2^. ' 








Z> A qp = 


cos 2 « sin 
sin 2 g; 


2 * 

— = COS*/, 






D<f,s 
A ~ 

r 


= \/( sin2 


9 ) +X 2COs2 ^) sec ^' 




A 


= v /(«n« 


B 2 \ 
9> + X* cos f >7 








=v / h( 1 


+ jj ) + 2 ^p 


— ljcos2 


a cos 2 X ; 


n ... 


Zfyssin;;, 


sin2«Z),j« 







^ ' i?sin qp .Z?sin qt. sin 2 qp ' 

r. sin 2aD-)S 

D,w— — -\— . 

A ' B sin qp sin 2 qp 

In the case of the prolate ellipsoid, the notation 
. „. 2(B - — A 2 )cos2a 



B*-\- A*-\-(B* — A*)cos2a' 
sin*} = sin i sin X, 



gives the equation 

5 = y/(Z? 2 cos 2 ce -\- A 2 sin 2 a) 8. ^. , 



— 157 — 

In the case of the oblate ellipsoid, the notation 

X = if 71 — X, 

. „., 2(A 2 — B 2 )cos2a 



B* + A 2 -\-(A 2 — B*)cos2a' 

shid' = sin i sin X' , 

gives the equation 

s = ^ (A 2 cos 2 a 4- £ 2 sm 2 a) %X'. 
In the case of the sphere the equations become 

B = A, i = i'=0, 

S = AX; 

and this result of this case is obtained by Bobillier. This case also 
gives the equation 

sin 2 « 



D xV = 



sin (f> sin 2 cp 

sin2« 



21\) 



(sin 2 a-|- cos 2 « sin 2 P.) y' (cos 2 « — cos 2 a sin 2 A) 

which by the notation 

cos2; // = tana, 

sin d" = sin i" sin A. , 
becomes 

T) 2secfl" 

* ^ sin « (1 + tan 2 »• sin 2 A) ' 

y = -4-gV,(tanV,Jl) 

T sin a v 7 ' 

= 2 sin a tan 2 a ! 3\„ ( — sec 2 a sin 2 &'", k) -J- 2 sin a 9v, X 

smi"tand' r cosl 



-f-2tan [ -^ 
300. Returning to the general case of the ellipsoid, let 



sin « 



— 158 — 

a and /? be the limiting values of (p for the upper portion of the 

curve, and 
a' and $' the limiting values for the lower portion ; and let 

q = i(a + 0), £ = !(/?— a), 
r!'=l(a'+F), e'=±(p' — a'). 

Hence the following values of M and N are obtained 

iV= £sin2a -J- .Msincc == -|sin2/3 -[- -Sfsin/?, 

— N= i sin 2 a' -f Jf sin a' = | sin 2 0' -f Jf sin (?, 

■nr cose cos 2 »/ cose' cos 2 >/ 

cos // cos if ' 

JY= tan^(cos 2 ij cos2e — cos 2 ecos2?]) 

= |-tan^(cos2e — cos2i;) = tan?}(cos 2 e — cos 2 *;) 

= |-tani;'(cos2?/ — cos2« / ) = tan?/ (cos 2 ?/ — cos 2 /) 

= tan i] sin a sin /? = — tan if sin a' sin [Y 

. , ' sin tj sin a sin @ — sin rf sin a' sin ^' 

y/ sin <p (cos 37 cos <jp — cos e cos 2 ?/) sin <p (cos ?/' cos cp — cos e' cos 2 ?/) 

s ^[ — (cos qp — cos a) (cos qp — cos /?) (cos (]p — cos a') (cos <p — cos ^')] 

s sin qp(cosg) -(- M) 

_ vT — (cos 2 cp — 2cos?/ cose cosqo ~\- cosk cos^) (cos 2 qp — 2cos//cos«'cosg9 -)- cos«'cos/3')] 

sin qp (cos qri -j- M) 

_ y/[sin 2 y (cos y -f ilf ) 2 — JV^ 2 ] 
sin q& (cos (jd -|- M) 

The numerators of the first and last values of cos*/ give, by 
direct comparison, 

— 2 M= cosa -f- cos/5 -[- cosa'-|- cos/j'= 2cos?j cose -|- 2cos?/cose', 

whence 

cos?jcos?/cose':= (cos2?j — cos 2 ?]) cose = — sin 2 ?; cose, 
cos ?; cos?/ cose = — sin 2 ?/ cose', 



— 159 — 
cos 2 »]cos 2 r/ — sin 2 ?] sin 2 •»/== cos(rj -|-i/)cos(j/ — rj) = 0, 

The comparison of the values of N, shows that the value of 1/ 
must be obtuse, whence 

cose' = tan fj cos £ , 
cos £ = — tan r[ cos e' . 

301. The general case of the surface of revolution admits of one 
integration, by denoting by v the ordinate of the meridian curve of 
revolution, which gives 

- = — = — DAogv, 

this equation, substituted in (144 29 ), gives, by integration, 

**" — *(* + *)> 
in which 

v is the ordinate of the meridian curve at the origin. 

This form of the equation is, however, limited to the case in 
which the curve has a point, in which its direction is horizontal. 
But every case is included in the form 

smS = 



in which Mis an arbitrary constant. 

302. In the case of the surf ace formed by the revolution of the equi- 
lateral hyperbola about its asymptote, which may be called the equilateral 
asymptotic hyperboloid, if the equation of the revolving hyperbola is 



— 160 — 
the equation (159v 4 ) becomes 

■ s _ M 

Silly/ jj, 

and, therefore, the inclination of the curve of this catenary to the arc of the 
meridian is constant. 

When M is greater than b 2 , the curve is impossible, but when 

M=±b 2 , 

the catenary becomes a horizontal circle, and 

S y ,=±±7l. 

303. It may be inferred from the comparison of the two pre- 
ceding sections, that, upon the circle of intersection of any surface of revo- 
lution with the equilateral asymptotic hyperboloid of equation (159 31 ), the arc 
of the catenary of either surface makes the same angle with the meridian 
curve of tliat surface. Hence, the limiting horizontal planes of the catenary 
of equation (159 16 ') are the intersections of the surface of revolution upon 
which it lies with the equilateral asymptotic hyperboloid, of which the equa- 
tion is 

v(!/-\-?/o) = v yv 

The catenary extends over that portion of surface which lies exterior to 
the asymptotic hyperboloid, and does not extend over that portion of surface 
which is included within the hyperboloid. 

304. To complete the solution of the catenary upon the equilateral 
asymptotic hyperboloid, the equation (159 31 ) gives 

tan,,/ = — -Ls v v == t — ; — rsj 

whence the following equations are obtained ; 

(j/+^o) 2 =^ 2 cot^/, 



— 1G1 — 



Dyj = — 



2(y + yo)sin s j; 



2 V J 



n h*-D„\p 

^r— 2(y + y„)sin 2 *, 



But it is found by § 285 that 



n sec g. tan;, (y -f- y„) tan g. 



v 6 2 cos^, ' 

whence 

7-i tan f,, 



2 sin 2 *, cob J. ' 



of which the integral 


is 


W = 


tan J. 


2 sin*. 



-|- tan *,log tan (^tt -j-l *, ) . 

305. If the chord is not strictly confined to the surface so as 
to be incapable of removal from it, but if it simply lies upon the 
surface, without the power of penetrating it, it must leave the sur- 
face whenever the pressure becomes negative, that is, when the 
sign of R, computed by (142 29 ), is reversed. The points at which 
the chord leaves the surface are, therefore, determined by the 
equation 

R = 0. 



>♦= 



21 



1G2 — 



CHAPTER IX. 

ACTION OF MOVING BODIES. 

CHARACTERISTIC FUNCTION. 

306. Related to the idea of the potential, and, in some 
respects including it, is that of the action of a system as proposed 
by Maupebtuis. Every moving body may be regarded as constantly 
expending an amount of action, equivalent to the power which its 
motion represents, that is, to the product of the force of the moving 
body multiplied by the space through which the body moves. 
Hence, with the notation of Chapters II. and III., if V designates 
the whole action expended by the system, the action expended at 
each instant is 

d V= 2Z l (m 1 v 1 ds 1 ), 

and the total expenditure of action is 

The function V is called by Hamilton the characteristic function 
of the moving system, and he has resolved the problem of dynamics 
into the investigation of its form and properties. 

307. If the power, with which a system is moving at any 
instant, is denoted by T, its expression becomes, by (4 20 ), 

The preceding expressions for the expended action give, there- 
fore, 



— 163 — 

D, V= S 1 (m^ D th ) = S 1 (n h v l a ) = 2 T, 
V=J2T. 

PRINCIPLE OF LIVING FORCES, OR LAW OF POWER. 

308. If 12 denotes that function which, in the case of the fixed 
forces of nature, is the potential of the moving system, its change 
for any instant is, by (3434) and § 58, 

d£2=dT=2[(m 1 F 1 df 1 ). 

Hence, in the case of the fixed forces of nature, if II is an arbi- 
trary constant, 

T=Q + H, 

which is only the analytical form of the proposition of § 58, and is 
called the principle of living forces. The term living force denotes the 
power of a system, so that this principle may, with equal propriety, 
be called the latv of power. 



CANONICAL FORMS OF THE DIFFERENTIAL EQUATIONS OF MOTION. 

309. The equation (8 15 ) may be written in the form 

d £2 = 2 1 (7n 1 D t v 1 d Sx) 

= D l ^ 1 (m 1 v 1 ds 1 ) — ^^m^d D t s x ) 

= D,2i(«i» 1 ^) — ^iOiM^i)- 

If, then, i; l5 ?j 2 , rj 3 , etc., are assumed to be the independent 

elements of position of the n bodies of the moving system, s 1} s 2 , etc., 
may be regarded as expressed in terms of these elements, so that 

v = Z> t s = ZJD J .sD t r } ). 



— 164 — 

With the notation 

D t r)=r)', 

this equation is resolved into the equations represented by 

The substitution of these values give, if T 1Urj , denotes T ex- 
pressed by means of rj 1} rj 2 , if 1} tf iy etc., 

Zi K *>i D n s x ) = S x {m 1 v x D 7f v x ) = D n , T Vt n , , 

Z 1 (m 1 v 1 I) n v 1 ) = D v T v>7] ,; 

whence 

I> v £2 = (D t B y -I) v )T V)V ,. 

This expression represents the elegant forms of the differential equa- 
tions of motion given by Lagrange ; but the mode of investigation is 
adopted from Hamilton. 

310. In the special case, in which the independent elements 
of position are the rectangular coordinates, x, y, z, of the different 
points of the system, these equations become 

v * = x'*+y'*-\-z'% 
D x , T Xj x , = mx' ■=■ mD t x, 

D X I2 = mD t x = mD* x. 

When the coordinates of the system are subject to conditions, 
these equations are still applicable, provided that the forces, by 
which the conditions are maintained, are included in the forces of 
12, or more properly of dfl. The values of D x £2 and D v £2 can be 
obtained from the given differential expression of i2, even when 



— 165 — 
such expression is incapable of integration ; for this form gives 

311. By means of the notation 

rj[, rf 2 , • • • • e tc, may be eliminated from the value of T, and 
T may denote the resulting value, expressed by means of iy x , 
r i2 , w a , o) 2 , . . . . etc. 

Since T is a homogeneous function of two dimensions in respect 
to rf x , if 2} etc., it satisfies the equation 

2T=Z v {ifD v ,T lhV ,) = 2 v {if W )- 

whence 

2 d T= Z v (to drf -f ifd w) . 

But the variation of T, derived by the usual method, is 

dT=2 v {D n T 7lj7l ,d>rir\-<odrf); 

which, subtracted from the previous value of 2 d T, leaves 

dT=2 v (r I 'd6> — D v T V)V rd> n ). 

This equation is equivalent to the two equations 

D T V}0 = rf, 
T) t — D T 

and Lagrange's canonical form assumes the following expression given ly 
Hamilton, 

D t w=I) v (£2— T V}0 ). 

312. But £2 is, in the case of the fixed forces of nature, a 
function of r a , ij 2 , etc., without other variables. If, then, in this case, 

tt — t n ■ 



— 166 — 

the preceding equations assume the simple form 

D t (o = — D V II V>0) , 

which are given by Hamilton, in which £2 may involve the time. 



VARIATIONS OF THE CHARACTERISTIC FUNCTION. 

313. The variation of the characteristic function, taken upon 
the hypothesis that the time does not vary, is 

dV=f2dT. 

But, from the preceding equations, 

dF^S^adij' + D^fdri) 

the sum of which and of the equation 

dT=d£2 + dH, 

is 

2d T= Z v (a)dii' -\- D t cadii) + d H 

The variation of the characteristic function is, therefore, 

d V= 2 n (atdi) — to di h ) -f id II, 

in which w and rj are the initial values of w and rj . If, then, V is 
expressed as a function of the initial and final coordinates, 17, w, rj , 



— 167 — 

and w , and of the constant II, its derivatives are 

D„ V= w , D Vo V= — M , 
D H V=t. 

By means of these equations, the problem is resolved by Hamilton into 
the determination of the single function V. 

314. In the case in which the independent elements of posi- 
tion are the rectangular coordinates, these equations become 

W = mx = mD t x = D x V, 
to = iiix'q = mD t x = — D X(j V. 

315. If the expression of the forces involves the velocities 
the final expression of d T in § 313 is incomplete, and the present 
mode of investigation is not easily and simply applicable to such 
cases, which is of less importance, because these cases are not, in 
the most comprehensive view of the subject, the cases of nature. 



PRINCIPLE OF LEAST ACTION. 

316. When, in the case of the fixed forces of nature, the ini- 
tial and final positions of the system are given as well as the initial 
power with which the system is moving, the variation of the charac- 
teristic function vanishes, and, therefore, the function is generally a 
maximum or a minimum. The action expended by the system, 
which is measured by this function, is also a maximum or a mini- 
mum ; or, in other words, the course by which the system is com- 
pelled to move from its initial to its final position through the 
action of the dynamic laws, is that in which the total expenditure 
of action is a maximum or a minimum. But it is obvious that, in 
most cases, and always, when the paths in which the various bodies 



— 168 — 

move are quite short, the described course cannot correspond to the 
maximum of expended action ; and, therefore, in most cases the sys- 
tem moves from its given initial to its given final position with the least possi- 
ble expenditure of action. 

Many examples can, however, be given, in which the expended 
action is, in some of its elements, a maximum ; although, even in 
these cases, the expenditure is a minimum at each instant, or for 
any sufficiently short portions of the paths of the bodies. 

317. This principle of least action was first deduced by Maupee- 
tuis, through an a priori argument from the general attributes of 
Deity, which he thought to demand the utmost economy in the use 
of the powers of nature, and to permit no needless expenditure or 
any waste of action. This grand proposition, which was announced 
by its illustrious author, with the seriousness and reverence of a 
true philosopher, is the more remarkable that, derived from purely 
metaphysical doctrines, and taken in combination with the law of 
power which likewise reposes directly upon a metaphysical basis, it 
leads, at once, to the usual form of the dynamical equations. 

318. To deduce the dynamical equations from the combina- 
tion of the principles of least action and living forces, add together 
the two variations of T, 

dT=dS2, 

= 2 n (D v T Vj n ' — D t m) d i] -|- D t Z n {tad i] ) . 

If the sum is introduced into the variation of V, the result, 
reduced by the condition that at the limits of integration, 

becomes 

d V= Z v f t {D v T n , n , — D,m -f D v S2)drj = 0. 



— 1G9 — 

The factor of drj, in this expression, must vanish by the princi- 
ples of the method of variations, which gives immediately the gen- 
eral expression of Lagrange's canonical forms. 



PRINCIPAL FUNCTION AND OTHER SIMILAR FUNCTIONS. 

319. The function S determined by the equation 

JS= V—Ht=J t {T+£2), 

is called by Hamilton the principal function, and its variation deduced 
from that of V is, obviously, 

djS=dV—t$H—ffit 

= 2 v (<odr) — w $?j ) — Hdt. 

Hence, if S is regarded as a function of rj, rj Q , a, w , etc., 

with the time t, its derivatives are 



D v S=o), D Vo S= — w , 
D t S= — H. 

The principal function may, therefore, be used in the same way ivith the 
characteristic function in the determination of the motion of the system. 

320. Many other functions, as suggested by Hamilton, can be 
substituted for the principal and characteristic functions. Thus the 
function 

gives 

=fz n (r } do>' + D fi nd<r } —I) v T n)0 dri) 

22 



Hence, 



— 170 — 
=fz v (r)da>' + D v T v> v ,drj) +fd£2 

=zS v {i]dbi — rj d w ) — td H. 

B a W= n , D ao W=- Vo , 
D H W= — t. 



321. The introduction of 

Q= W+tH=f t (2: n (f i a>') + H), 
gives, in like manner, 

322. Other functions can be formed by the combination of V 
and W, or JS and Q. The combination may be such that for some 
of the coordinates, the function shall have the same form as V or S, 
while for the remaining coordinates it shall have the form of W or 
Q, and the function 

U=V'—W", 
or 

P = jS'—Q", 

can be substituted for V or S. 



171 — 



PARTIAL DIFFERENTIAL EQUATIONS FOR THE DETERMINATION OF THE CHARACTER- 
ISTIC, PRINCIPAL, AND OTHER FUNCTIONS OF THE SAME CLASS. 

323. By substituting in the equation 

for 1], a), etc., as well as for t and H, their equivalent expressions, as 
partial derivatives of V, 8, W, Q, U, and P, partial differential equa- 
tions are obtained, the integrals of which give the values of these 
functions. To facilitate the expression of this substitution, T and £2 
may be assumed to have such functional significations that 

£2(t,t}) = £2. 
The partial differential equations are, then, 

T{ n ,D n V) = n(D H V, n ) + H, 
T{r ] ,D Tj S) = a — D t S, 
T{D U W,to) = 12 (- D H W,D U W) + H, 
T(D a Q,a>) = £2(t,D u Q) + l) t Q. 

324. When the independent elements of position are the rec- 
tangular coordinates of the bodies, these equations become, by the 
notation of (54 31 ), 

S -(k nJ $ = 2£ *(D H V,z) + 2H, 

Z m m(z' 2 -\-?/ ,2 + /*) = 2f2(—I) H W,± i D x ,w) + 21I. 



— 172 — 

325. Through the preceding investigations, the forms are 
developed by which every dynamical problem can be expressed in 
differential equations. It only remains, therefore, before applying 
these forms to especial problems, to consider those methods of inte- 
gration which are best adapted to their discussion. 



CHAPTER X. 
INTEGRATION OF THE DIFFERENTIAL EQUATIONS OF MOTION. 

326. In discussing the differential equations of motion, it 
might be permitted to suppose a previous knowledge of all that has 
been written upon the integral calculus. But since the profound 
philosophical views, with which this subject has been illuminated by 
Jacobi, have not yet passed from the original memoirs into the text- 
books, a development of them is required by the plan of the present 
work to facilitate its further progress. 

I. 

DETERMINANTS AND FUNCTIONAL DETERMINANTS. 

327. If (?z-)-i) 2 different quantities are given, which are 
represented by 



— 173 — 

in which every number from to n can be substituted for Jc or for 
the number of accents denoted by i ; and if all possible products of 
(n -J- 1) factors are formed similar to 

±aa[d-l a [ :\ 

in each of which the same number is never repeated, either for k or 
for i; and if these products are successively formed by mutually 
interchanging two of the inferior numbers, and at the same time 
reversing the sign of the product ; the sum of the products has been 
called by Gauss the determinant of the given quantities, and may be repre- 
sented by 

%> n = 2 ±ad x di a« 

Thus, for example, 

^0 = -2" + a = a, 

§&! = IE + aa\ = aa\ — a x d, 

^2 = ^ i a<h a i — ad^a'l — aa' 2 a" -\- a 1 a' 2 d / 

— a x a a 2 -J- a 2 a a x — a 2 a x a . 

The same result might also have been produced by mutually 
interchanging the accents without disturbing the inferior numbers. 

328. The sign of the determinant would be reversed, by 
reversing the sign of the product originally assumed as the basis of 
the subsequent changes. 

329. If, for the different values of Jc, all the given quantities 
are equal, so that 

the determinate vanishes. For, by interchanging Jc and Jc in all the 
terms, the sign of the determinant is reversed by the regular process 
of formation, whereas if Jc is substituted for Jc' and the reverse, no 



— 174 — 

change is produced on account of the equality of the given terms. 
Hence 

n nl 

or 

330. Whenever all those values of the given elements vanish, 
for which i is as great as m, while h is less than m, which condition 
may be denoted by the equation 

the form of the determinant may be simplified. For it is evident 
from inspection of the fundamental product, 

(««x «irr 1 i) )(« ( r«a i) «<">)> 

regarded as separated into two factors, that every elementary prod- 
uct, produced by an interchange between the inferior numbers, such 
as to transfer one of these numbers into the second factor, vanishes, 
and may be neglected. Hence 

<& n =.2± aaWi a%-v.Z ± a™a%ff a« 

if 

6JL -V_J_^(»n) (m + l) (n) 

uv >m,n ^ I a m a m + l U n • 

331. When, in the preceding proposition, m is equal to n, so 



that 






a k<n 0, 


it becomes 






. ^=^_ lfl W; 


for, in this case, 






% hn = ^±a^ = a^. 



— 175 — 

332. When, in addition to the preceding equation, the values 
of the elements vanish, for which m is equal to n — 1, so that 

U>n-l) A 

«i<»i — 1 U J 

the value of the determinant becomes 

333. Whenever the equation (174 10 ) is true for all values of 
m, it may be written in the form 

»P*>=0; 

and the determinant is reduced to the single term 

^ n =aaW2 < n) - 

334. If a determinant is formed from the given elements, with 
the omission of all those of which the number of accents is i, and 
those of which the inferior number is k, so that n is the number of 
factors of each elementary product, this determinant is the factor of 
af in the expression of the determinant £&„. If, therefore, this par- 
tial determinaitt is denoted by okf, the expression of the complete 
determinant is 

The derivative of this expression is 
whence 

335. The preceding notation gives 

Qk = %,^ n =Z± a [4 a?>. 

Hence the expression of (Mp can be deduced from that of o4 



— 176 — 

by putting 

i = Je = j 

and those of — o/k (f) , or of — Q%, are deduced from that of ok by 
putting 

i = , or # = . 

336. If, in the third member of (175 21 ), «$ is substituted for 
ajj. l) , the expression of the determinant is that which corresponds to 
the case of § 329. Hence, 

and, in the same way, 

337. If a partial determinant is formed from the elements of 
Q&jjp, with the omission of those in which the number of accents is i', 
and those of which the inferior number is k', this determinant, taken 
with its proper sign, is the factor of a ( p in the value of okff. If, 
then, it is denoted by Q^p f the value of Q/bj(p is 

in which it must be observed that, from the definition 
These equations give 

= ^ y (ai?al?>i>.BP.i>aP».) = 2*,„ (aLW2> a p^aj. 

338. All the given elements which have k or k' for their infe- 
rior number, are excluded from the value of o^-tV? an d? therefore, 



I I 

this parti.il determinant is not affected by the interchange of Jc and 
1c , by which the terms of the complete determinant, comprehended 
in 

are transformed into those comprehended in 

But this last aggregate of terms is also represented by 
Hence these partial determinants satisfy the equations 

<*iy? = — <*&,*2 = — <*&? = <*i?# 

The determinant may, therefore, be written in the form 

= ^>»(Q^P(aiPaP - aJT>aS3) ). 

339. The solution of linear algebraic equations is easily 
accomplished by the aid of determinants. For if the given 
(n -f- 1) equations are 

u = at-\-a 1 t l -\- -j- aj n = Z k {a k t k ) , 

u'=a't 4~ «i4 + + a'Jn = ^k(a'Jk), 



v 



„m = a^t + fl f ^ + + ct { :H n = Z k (a["H k ) ; 

the sum, obtained by adding the products of the given equations, 
multiplied respectively by ok k , ok k , °^i n) , is 

<3l„4 = oA kU _}_ ok' k% { -f oi k n hi {n) = ^(q^V'') 

23 



— 178 — 

340. If all the quantities u, u', u", etc., vanish, t, t 1} t 2 , etc., must 
likewise vanish, unless the determinant vanishes. If, therefore, 
either of the quantities t, t 1} t 2 , etc., does not vanish, when u, u', u", 
etc., vanish, the determinant must also vanish, whence the equation 
(176 13 ) applies even when 

i' = i, 

or for all values of i' 

Hence, it is evident that 

t-.t^.U \t n = Qk^:okf:ok>'i> : q4,w 

341. The process, by which the value of i n was obtained, may 

be regarded as designed to eliminate the n quantities t,t 1 ,t 2 

t n _ 1 from the given equations. By precisely a similar process, the 

m quantities 4 h-, h C— i m &y be eliminated from the first 

m — |— 1 of the given equations, and the form of the resulting equa- 
tion must be 

Bu + 5V+ + J?(->«(-)= CJ m + C m+1 t m+1 + + CJ n , 

in which 

In the same wav, if 

r n — 2 + ok oA{ ok'l o&w 

r = 2 + ok im) o4 (m + 1) oifcw 

' m,n — " _i_ m «i -+- 1 n 5 

the quantities ti {m + l) , u (m+2) , u {n) may be eliminated from those 

of the equations (177 30 ) which give the values of t m , t m + 1 , t n , 



— 179 — 
and the form of the resulting equation is 

En + E'u' + + J5Wt»<-> = FJ m + 1L +1 C+i + + «*> 

in which 

But the two equations, obtained by these processes, must be 
identical in the ratios of their coefficients. Hence 

Em — Fm , 
or 

™Bm- 1 r m, n ?k<?i r m, n 



Vn' m + l,n <->\Dn 'm + l t n 



or by extending the series of ratios to all values of m, 

*«: «,: : ^-1 = ^1,.: «Sr M : : 0L> n ,„. 

But it is easily seen that 

r^.= ofeM = «._!, 
and 



whence 



ri > n ok> rn ' 3 



^n = ^ Qk r n = a^l-\ 



A repetition of the same process, in a different order, upon the 
given equations gives 

Hence 



— 180 — 

342. The ratio of the values of r n and i\ n may be prefixed to 
the series of ratios of (179 16 ) in the form 

The series of ratios gives, then, 
or 

This investigation is derived from Jacobi. 

343. The variation of a function of the quantities represented 
by a { k ] is expressed by the formula 

If, then, the values of the quantities, denoted by u {i \ are such 
that 

and if the corresponding values of t,t x , t n are denoted by 

t (k) ,t\, t ( n\ the expression of iff assumes the form 

fflt^J*) = S t [Daf ®» n (da? + (i, *))] , 
and therefore 

344. If the given quantities are such that 



181 



it is 


readily perceived that 








c*jp= 


Qkf\ 






Qkf{i,k) = 


— Qi?>(*, 


.*), 


and 










— *«* — ot 


1 = (T log. 


n ? 



which is given by Jacobi. 

345. A system of equations, similar to those of § 339, repre- 
sented by the form 

gives, in the same way, 

If 
an equation similar to (180 26 ) is derived, 

346. Let the («-}-l) 2 quantities, represented by c[ i] , be 
derived from the given elements a k i] and b ( l ] by the formula 

c k V) = aA J r<K J r aPW = 2 m {aWp), 

and let the determinant of these quantities be 

Q^ m =S±e^4 of. 

If only one term is taken in each of the quantities c { p, the 
general term of £&„ is represented by 

±a^aY ) a^' ) b^b^b^P 



— 182 — 

A mutual interchange of the letters Jc, followed by a mutual 
interchange of the letters i in the resulting terms, produces all the 
terms of Sh n , which correspond to the same combination (M) of 
accents m, m 1 , etc. A different combination of accents gives a dif- 
ferent set of terms ; and if 



Ml 



^ = 2 ± a^a^air^ <, 

g»w = 2 ±b {m) b { r' ) ^i n " ) Kt ] , 

denote the determinants of the given elements corresponding to one 
of these combinations, the complete determinant is expressed by 

which is given by Jacobi. 
347. In the case of 

p = n , 

there is only one combination (M) of the accents, so that in this 



case 



&.= fc.<3»., 



which was given by Cauchy. 
When 

P<n, 

there is no combination (M), in which all the accents are different 
from each other, and, therefore, it follows from § 329 that, in this case 

and that, in all cases, the combination (31) must consist of accents 
which differ from each other. 
348. In the special case of 

a k == V k ■> 



— 183 — 

which gives 

„(/) — r {k) 

the value of the determinant is reduced to 

G)„ >r /uJJ u (M)\2 

=-*>» *^M\ M n ) 3 



which, when 
is reduced to 



p-=n 



Qo =<m 2 



FUNCTIONAL DETERMINANTS. 

349. If the given elements a k i] are the derivatives of [n -J- 1) 
functions f,f 1} /„ of (n -f- 1) variables x,x\, x n , so that 

the determinant of the elements is called the functional determinant of 
the given functions. Thus, in the present case, all the terms of the 
determinant 

®°n = 2 ± D x fD Xifx D x J 2 D Xnfn} 

are obtained either by a mutual interchange of the variables, or by 
a mutual interchange of the functions, the interchange being 
accompanied in either case with a reversal of the sign, precisely as 
in deducing the terms of the ordinary determinant. The proposi- 
tions, which have already been given in reference to determinants, 
are easily applied to functional determinants. 

350. In the case in which all the functions, above the 
(m -\- l)st, are free from the first m variables, the condition of (174 9 ) 
is satisfied, so that the notation of (1742a) gives the equation (174 20 ) 



— 184 — 

351. In the case in which every function is free from the 
variables of which the inferior number is less than that of the 
function itself, the equation (175 10 ) is satisfied, and the functional 
determinant, reduced to a single term, is 

352. If the given functions are not independent of each other, the 
determinant vanishes. For if the equation, which denotes their mutual 
dependence, is expressed by 

77=0, 

its derivatives, with regard to the given variables, are represented 
by the equation 

^(^77^/ & ) = 0. 

The equations, included in this form, are identical with the 
linear equations of § 339 when the values of u vanish and 

t k = Df h n. 

All these values of t cannot vanish, because the equation, which 
expresses the mutual dependence of the functions, must involve one 
or more of them ; and, therefore, the determinant must vanish 
by § 340. 

353. If either of the given functions (f) contains any of the other 
functions, these functions may be regarded as constant in finding the 
functional determinant. For each derivative of f is the sum of two 

parts, one of which is derived by direct differentiation with refer- 
ence to the variable explicitly contained in the function, and the 
other part is obtained by indirect differentiation through the 
functions involved in f. The whole determinant may then be 
regarded as composed of two such portions. But the portion of the 
determinant obtained by the indirect differentiation off is the 



— 185 — 

same as if/,, not containing explicitly any variables, were simply 
a function of the other functions* This portion must, therefore, 
vanish, and the remaining portion of the determinant is that which 
is obtained by direct differentiation, conducted as if the functions, 
involved in f i} were constant. 

This proposition is applicable even where several of the given 
functions contain the remaining functions ; but not when they 
mutually involve each other. 

354. If the second of the given functions contains the first, 
if the third contains the first and second functions, and if, in general, 
each function contains all the previous functions, the preceding 
proposition is applicable. Hence if, by means of the first function, 
the first variable is eliminated from all the other functions ; if, by 
means of the second function thus reduced, the second variable is 
eliminated from all the subsequent functions ; and if this process is 
continued until each function is liberated from all the variables 
designated by an inferior number, although it may involve all the 
preceding functions ; the determinant is reduced to a single term as 
in § 351. This will often afford a convenient method of obtaining 
the functional determinant. 

355. In performing the successive eliminations, the operation 
must not be restricted to any prescribed order of the variables, but 
one of the variables, remaining in f, must occupy the place of x L . 
Hence there is not one of the factors of the determinant in the 
form of § 351 which vanishes, unless a function be obtained from 
which all the variables are explicitly eliminated, or, in other words, 
unless one of the given functions is included in the others and can 
be derived from them, so that they are not independent of each 
other. If, therefore, the given functions are mutually independent, their 

functional determinant does not vanish. 

356. If F, F 1} F n are given functions of /,/i, f p , 

24 



— 186 — 

which are themselves functions of the variables x, x 1 , x n , the 

derivatives of the functions {F { ) with respect to the variables (^) 
are represented by the equation 

n Xi F k = Z m {Dj m F k D rifm ). 

This equation coincides with (I8I24), if the notation for af is 
combined with the notation 

cf = D Xi F k , 
h% = » fm F k , 

The remaining notation and conclusions of §§ 346 and 347 
may, therefore, be applied to this case. Hence, by (182 18 ) the 
functional determinant of the independent functions (F { ), taken ivith respect 
to the same number of variables (x { ), which enter into (F t ) only as they are 
involved in the same number of independent functions (f) explicitly involved 
in (Fi), is obtained by multiplying the functional determinant of (F { ) tahen 
ivith respect to (f) by the functional determinant of (/,-) taken ivith respect 

to (Xi). 

If the number (j) -\- 1) of functions (f) exceeds the number (n -f- 1) 
of functions (Fi), the complete functional determinant of (F { ) is by (182 n ) 
the sum of all the partial determinants of (F { ) obtained by every possible 
combination of (n-\- 1) of the functions (f). 

If the number of functions (f) is less than that of the functions (F { ), 
the functional determinant vanishes, as in (182 25 ), tvhich corresponds to the 
proposition that the number of independent functions cannot exceed the num- 
ber of variables, by which they may be expressed. 

357. In the case, in which 

all the derivatives of (F t ) with reference to the variables (x t ) vanish, 



— 187 — 

except those included in the form 

D XiFi = D XiXi = l. 

In this case, therefore, 

V. = JE±D f *l> A * !>,.*., 

is the functional determinant of (a:,) regarded as functions of (f), 
and the equation (182 18 ) becomes 

&„ = 1 = #„&., 

or the functional determinant of (#,-) taken with respect to (f) is the recipro- 
cal of the functional determinant of (/,■) taken ivith respect to (#;). 

358. If in the linear equations of § 339, the values of (t) are 
expressed by the formula 

either of the equations is represented by 

u m = 2 k {D Xt f m D fiXk ) = D A f m = . 

unless 

m = i, 
in which case 

^ = ^ = 1, 

This value substituted in (177 3l) ) gives 

«.*/,** = A? «. = <*if>. 

359. If it is again assumed that 



— 188 — 

the equations of § 345 give 

v k = Z m {aFH% ) = Z m {D Xk f m D f J fi ) = D x J fi 

2 h W = ZtWfk = ^ = * log &.. 

360. By the same process, it may be proved that, if (fi) are 
the variables and (^) the independent functions, 

2»V, = * lQ g ^ = — * lo s ®w- 

But it must be observed, that in finding the derivatives of 
dx k they are supposed to be expressed as functions of the original 
variables, precisely as in the preceding section the values of d f h 
are supposed to be expressed in terms of f k . 

361. The equation (188 9 ) reduced to the form 

may be added to the identical equation 

The sum is, by (187 27 ), 

= 2 kji D Xk {^ n D fiXk § fi ) 

362. In the case, in which the arbitrary variation d is assumed 
such that 

except for the value 

i=0, 



— 180 — 
the preceding equation becomes 

If this equation is multiplied by / and added to the equation 

the sum is 

363. If the equations, by which the functions (/,) depend 
upon the variables {x t ), are represented by 

^ = 0, 

their derivatives are represented by 

i x *"»i\ J m J -k ijmj' 

The comparison of this equation with (186 4 ) indicates that the 
concluding propositions of § 356 may be applied to this case, pro- 
vided the negative sign is introduced as a factor of all the deriva- 
tives taken with respect to (/ £ ). Hence, if the number of the 
functions (f { ) is the same with that of the variables (a?*) , 

2 ± D X FB S F X D,J m = (-) n+1( & n Z + D f FD fiFl \F n , 

and 

6ft. / \» + i 2+Dx FD'i Fi -D* n F n 

n ~^ > Z±DjFDf x F x D/ n F n ' 

364. If the number of the functions (/ f ) exceeds that of the 
variables (x t ) and is p -j- 1 instead of n -f- 1, let (Fj) be the form of 
(Fj) when the last p — n of the functions (/,) are eliminated from 
it by means of the last p — n of the given equations. In this case 



— 190 — 
it follows from the reasoning: of S 354 that 



o 



= -2" + D F X D X F} D x f 1 s + Df F ,,!>/ F , „ -0/ F , 

2 ± D f FDf x F x JD fp F p 

But the equation (189 26 ) is applicable to this case if (7^) is 
changed to {F}), and, therefore, the introduction of a common factor 
into the terms of (189 26 ) gives, by means of the preceding equations, 

05 _ / y, + i 2+D^D^ F 1 D* H F n Df n+l F n+1 

^»— V ) 2±D f FD/ l F 1 D/ p F p 

365. There are various interesting and instructive relations 
between the partial determinants of functions which have been 
developed by Jacobi, and which will be found useful in discussing the 
theory of differential equations. If the number of the functions 
(fi) as well as of the variables (x t ) is increased to m -J- n -J- i, let 



') = 2 + D f-D* A D* f i ^ f ... 



If, then, from the function (f n+i ), all the variables x, x x , 

x n _ 1 are eliminated, and the functions f,fi f n -\ introduced 

in their places, and the function (f„+i) thus transformed is denoted 
by (fl+i), the values of 2B become 

k ^^K — 1 n + kJ n + i • 

The determinant of the (m -f- 1) 2 functions (Bff) is, con- 
sequently, 

S + Q&Q&'MZ ^^=6g, m+ ^2-\-D x f 1 D x fi, , D x fi. . 

^" J_ 1 2 ^ m <Jton — 1 -" J_ „J n n + lJ » + l n + m ./ »-)-»i • 



— 191 — 
But it is obvious that 

^n + m ^rt — 1— X J n + mJ n + m> 

whence 

^±a«$ »a ,, =-«-i«.+». 

3G6. If *$$ denotes the value which ( ^°„_ 1 assumes when all 
the derivatives relatively to x t are changed into the derivatives rela- 
tively to x n + k , it is evidently the factor of D x .f n+ . in the value of 
— SBjjp. In the value, therefore, of the determinant 

■2 + SBSBj 2BW, 

the factor of I? x /,A,/»+i A /„+„, is 

(_)»+i^-j : <e<igJ <®<w. 

But the factor of the same quantity in ^ n+m is, by inspection, 

( ) m + 1 ^-\-D x fD x f, . .Dr f D x f .... A: f , 

V / _!_ m + lJ m + 2^1 m+»^ii-r 

It, therefore, follows from (191 5 ) that 

2 ± 2B SB} SBW 

V ; n + 1"* - J_ m + 1/ m + 2/1 m+n J n — 1" 

367. The factor of ^_J nJri in the value of 0Bj? is — ^jf" 1 ', 
and therefore the determinant 

(— )-^ + gB<@ ^ <8£-« 



— 192 — 

is in (191 12 ) the factor of D x f n+1 D \f n +2, Dx m -J n + m > But the 

factor of this same quantity in §&> n+m is, by inspection, 

( f^ + A fDx f, . D x f ,D X f D x f 

\ J — " J_ n + lJ n + lJ 1 n + m J m — 1 m J m „ J n 

= C )"l»+l) 2 + Dx fD x f, ...... Dx f. 

V ) -" * _!_ mJ m + lJl m + nJ n' 

Hence it follows from (191 5 ) that 

2±®<® 1 % ^ _1) 

— (_V + 1 <m« . 2 + Dx fD x /. D x f . 

368. By the same process it will be found that, in general, 

;s + SB SB;®? 0fc 1)C @/^ +1 ( ^- 1} 

— ( V' + '6j5"' ^ + -0* fDx f Dx f ,. . 

369. If the factor of S^ in the value of (191 29 ) is denoted by 
( — yk k , this expression gives 



2 + <&%% «a-« = ^(*A) 



in which neither the quantities ( ( ® k i] ), nor any function of them, such 
as X k , contain the derivatives of/„. Hence the derivative of /„, with 
respect to x n+k , only occurs in this expression because it is in ^ k , in 
which its coefficient is ^ B _!, so that the term of the preceding 
expression which contains this derivative is X k %> n _ 1 D x h f n . If fj, k 
is the coefficient of the same derivative in 

2 + Dx fDx f. ... D x f 
the equation (192 8 ) gives 



— 193 — 
The comparison of (192 9 ) with this equation gives 

It is to be observed that, from their definitions, the functions 
fi k and 2B A . are both of them partial determinants of the same 
functions /, /i, • • . ./„_i the former being taken with respect to 
the variables x m , x m + 1 .... x n + m excluding x n+k , and the latter 
being taken with respect to the variables x, x 1 . . . . x n _ 1 and x )l+k . 

In the case, therefore, in which m and n are equal, these 
two determinants are formed with respect to an entirely different 
set of variables, and each of the variables x n+k is taken in succession 

from the set x n , x n+1 x 2n in forming \i, k and combined with the 

set x, x x x n _ x , in forming 2B A . 

370. The first member of (193 3 ) does not contain any deriva- 
tive of /„ with respect to a variable of which the inferior number 
is less than m. The factor, therefore, of such a derivative as D x f n 
in the second member vanishes identically ; which is represented 
by the equation 

2 t foZ ± D* n+k fD*J x D x J % ^ n _ 1 /„_ 1 )= 0. 

371. If in the equation (191 3 ) 

n = 1, 
this equation becomes, by writing n — 1 for in, 

%> n =^2±B x J\D xJ l D xJ l 



But 



25 



— 194 — 

so that if x is supposed to be a function of the other variables and 
/ to be equal to x, these equations are reduced to 

z k (oA k n.J) = z k {^ k D. kX ) = z±D x j\n xJ i b x ji 

= Qk + £ k {cA k n, l x); 

1 

in which 

and, by (176 3 ), — ok k is deduced from ok> by changing the deriva- 
tives relating to x k into the derivatives relatively to x. This 
equation is derived from Lagrange. 

372. In the greater portion of these formulas upon functional 
determinants, the derivative taken with regard to either of the 
variables may be supposed to be frequently repeated, so that D x 
may be substituted for D x , and h may even be zero. Thus if, in 
§ 365, D x is substituted for D x , and if 



n = l 



the equations of that section are reduced to 

=f D *Ji-fi D *J 



Hence if 



—f2J) A 



*' = 7 



and if n is written for m-\-l, the equation (190 31 ) becomes 



— 195 — 

If each of the functions (/•) is multiplied by t, the values of 
the functions (/,) remain unchanged, and therefore the value of the 
determinant 

is multiplied by t n + l . 

373. A system of functions (/•) can always be found such that 
their determinant, with respect to the variables (#,-), may be equal 
to a given function IT of those variables. For, if all these functions 
except /„ are assumed at pleasure, and if f\ represents the form 
of /„ when all the variables except x n are eliminated and the 
remaining functions (/•) are introduced in their place, the required 
determinant becomes 

«.■=&*_!#.. /i = J7. 

Hence,/*, is by (187 10 ) determined by the integration 

in which it must be observed that the quantity under the sign of 

integration is expressed in terms of/,/! f n -i an d x n . 

In the case of 

77=1 
this formula becomes 



/i =/.«£. =/».-. 



The substance of all these investigations upon determinants is 
taken without important modifications from Jacobi. 



— 196 



MULTIPLE DERIVATIVES AND INTEGRALS. 

374. The functional determinant is shown by Jacobi to be of 
singular use in the transformation of multiple derivatives and inte- 
grals. The expression of these functions is facilitated by the 
notation 



and 



J)n-m+l__ 

Jm • • ' • 


VAnSm + l 


f*n — m + l 

1 ■ ■ 


Pn — m-\-l 


J m " ' ' ' 


Jfm/m+l 



'Jni 
•Jn 



If then 



£2 = D}+] W, 



a new variable x n , which is a given function of all the variables, f { 
may be substituted for either of them as /„ in W, and the new 
derivative is given'by the formula 

" S2B xJn = I)} B Xn W. 

Another new variable x n _ x may next be introduced instead of 
/ n _i in the same way, and this process may be repeated of substi- 
tuting successively for each variable f t a new variable w i} which 
shall be a function of all the other variables remaining in the 
derivative at the instant of the substitution of x { , until, finally, an 
entirely new set of variables shall be introduced into the derivative. 
The final form is 

^D x fD xJl D.rJ n = D'^W. 

From the comparison of this form with § 351, it appears that 



— 197 — 

the factor of £2 is identical with the determinant of that section. 
From the reasoning of §§ 353 and 354, it follows that the determi- 
nant is not changed by substituting in either of the quantities (/ ; ) 
regarded as functions of the variables (#,•) the values of any or all 
the preceding functions in terms of these variables. But each of 
the functions (/,-) contains, in its present form, none of the succeed- 
ing functions ; so that, after this substitution, it is expressed in terms 
of (.r,) . Hence 

375. The preceding equation gives, for the multiple integral 

in which the limiting values of (a? t -) may be supposed to be constant, 
while those of (f t ) may not be constant. If then IT is determined 
by the integration 

n=f f n, 

so as to contain neither of the variables (#;) except as they are 
involved in (/,), it is by § 353 unnecessary to have regard to the 
derivatives of IT otherwise than as they are dependent upon / in 
finding the value of the determinant, which is the first member of 
the following equation, and which therefore becomes 

s+n.n^fi^fi D xJn = %, n D f n= % v a. 

But by (189 8 ) 

Z±D x TTD Xifl D x j 2 D Xnfn = v^.(/7o4,); 



— 198 — 
and, therefore, 

//.!; fl =*J.^S(^A)=J?./;:. M ,. i+1 ....Km s (7r a k i ), 

in which lim x denotes that the function to which it is prefixed is 
referred to the limiting values of x k , so that the difference of the 
values of the function at these two limits is represented by this 
notation. 

But since 

it is evident from (197 13 ) that 



£ \im x {IfQk) = lim x J f n II; 



and a similar equation may be given for each of the terms of the 
last member of (198 3 ), whereby this equation is reduced to 






The multiple integral of the (n -\- 1) th order is thus reduced to 
2ra -f- 2 multiple integrals of the wth order, and this reduction may 
be continued until the whole process is made to depend, upon single 
integrals, of which one is performed with reference to /, and the 
number, performed with reference to any other of the variables (/;), 
is 

2 l (n-\-l)n (n + 2 — i). 



— 199 



II. 

SIMULTANEOUS DIFFERENTIAL EQUATIONS AND LINEAR PARTIAL DIFFERENTIAL 
EQUATIONS OF THE FIRST ORDER. 

37C An equation 

/=0, 

of which the derivative vanishes identically, by means of the 
simultaneous differential equations represented by 

in which (X,) are given functions of the variables (#,), is called an 
integral of these equations. It is a general integral if it involves arbi- 
trary constants, and a particular integral if it does not involve arbi- 
trary constants. When it involves an arbitrary constant, it is more 
conveniently expressed in the form 

f=a, 

in which a is an arbitrary constant. 

377. A function /, which satisfies the linear partial differential 
equation of the first order 

is called a solution of this equation. By means of the notation 

r. = i,(JrA) 



' m,n -~"(\,-' 1 j i) " 



— 200 — 
this equation may be written 

^./='0. 

378. The first member of every integral, expressed in the form (199 8 ) 
or (199 19 ), of the simultaneous differential equations (199 n ) is a solution of 
the partial differential equation (200 2 ) ; and, conversely, every solution of 
the partial differential equation (200 2 ) is the first member of an integral of 
the simultaneous differential equations (199 8 ), and its second member is any 
constant. For the derivative of (199 8 ) or (199 19 ) vanishes by the 
substitution of (199 n ), which gives 

2 t (l>. l fDx t ) = r n f=0, 

that is, / satisfies the equation (200 2 ). Reciprocally, the satisfying 
of this condition is all that is required in order that (199 19 ) may 
be an integral of (199 8 ). 

379. If the equation (199 8 ) is solved relatively to x, so as to 
express x as a ftfnction of the other variables (#,-), the equation 
(199 25 ) becomes 

x— r 1 , n x = o, 

which is distinguished from (200 2 ), because the function x, of which 
the derivatives are taken, is involved in the functions (-X*), whereas 
/ is not involved in these functions. 

380. A solution of (200 2 ) which shall, for a given equation 
between the variables, become equal to a given function, may be 
determined by means of series. For this purpose, let the given 
equation be 

t = r, 
in which t is constant, and t a function of the variables, and let the 



— 201 — 

solution become a function c/> of the variables when this equation is 
satisfied. If then t were assumed to be also one of the variables 
of the given equation, and such that in forming the simultaneous 
equations 

Dt—1; 

by which the simultaneous equations become 

D l x i = X i ; 
and the given partial differential equation is 

assume the functional notation 

□ =■—//., 

and the integral of the partial differential equation with reference 
to t is 



/— 9— D/=0, 



which gives 



(1 — D)/=y» 



This value of/ taken from Cauchy, expresses a true solution of 
the given equation if □'(p is finite for all values of i and vanishes 
when i is infinite, which is always the case for sufficiently small 
values of t — t . 

381. There are n independent solutions of the partial differential 
equation (200 2 ) and no more than n independent solutions. 

26 



— 202 — 

First. The equation (200 2 ) has n independent solutions. It 
has been proved in the preceding section that it has one such solu- 
tion. Let it then be assumed that m such independent solutions 

have been obtained, denoted by f n , f n -\ /»-m+i- These 

independent solutions may be substituted for the m variables, 

x n , x ll _ 1 x n _ m+1 , with regard to which they are independent ; 

and if f X:f denotes the value of/ when expressed in terms of the 
new variables, the equations of substitution are represented by 

n — m -\- 1 

But since 
the substitution of these equations in (200 2 ) reduces it to 

r 

in which the functions f k may be regarded as constant. This 
reduced equation has, then, a solution by the preceding section ; 

its solution does not involve the variables x n , x n _\ x n _ m+l , 

and is independent of the given m solutions. The given equation is 
then proved to have another solution independent of the given 
solutions ; and this number may again be increased by the same 
process, until the n independent solutions are obtained, fi,f 2 • . • • ./„. 
Secondly. The equation (200 2 ) cannot have more than n inde- 
pendent solutions. For if there are (n -\- 1) solutions (/,), each 
gives an equation represented by 

which may be regarded as a linear equation between the quantities 



— 203 — 

(X;). By the usual process of elimination, if $% denotes the 
functional determinant of (/,) with respect to the variables (#<), 
these equations give, by § 340, 

But all the quantities X t do not vanish, and, therefore, 

^„ = o, 

or the (n -f- 1) functions (f) are, by § 355, not independent of each 
other. 

382. It is evident, from the preceding demonstration, that any 
function of the solutions of the linear partial differential equation (200 2 ) is 
itself a solution of that equation. 

383. A system of finite equations, of which the derivatives 
are satisfied by the simultaneous equations (199 12 ), is called a system 
of integral equations of the simultaneous differential equations. This system 
is said to be general, when, by the successive elimination of the con- 
stants, it can be reduced to a form, in which each equation involves 
an arbitrary constant not included in the other equations, and it is 
complete when the number of finite equations is equal to that of the 
given differential equations. When reduced in the method just 
proposed, the general system is represented by 

in which the functions (</>,) are independent of the arbitrary con- 
stants (pi). The particular system is represented by a set of similar 
equations, combined with other equations, which involve no arbitrary 
constants, and which are represented by 



— 204 — 

384. Each equation of a general system of integral equations, 
reduced to the form (203 24 ), is an integral of the given simultaneous differ- 
ential equations. For the derivative of (203 24 ), when reduced to a 
finite equation by the substitution of the given differential equations, 
is independent of the arbitrary constants (/?;), and vanishes, there- 
fore, independently of the equations themselves in which these con- 
stants are involved. When the system is general, therefore, the 
functions (^,) are functions of the solutions (f) of the partial differ- 
ential equation (200 2 ). 

385. If the system is particular, and if the number of the 
equations (203 31 ), which are free from arbitrary constants is m — n, 
the same number of variables can be eliminated, by their aid, from 
the functions (X t ) and (<p,). The equations, to which (203 24 ) are 
thus reduced, are integrals of the simultaneous differential equations, 
represented by 

Dxi = Xi, 

in which the variables (%?), of which the number is m, are those 
which are not eliminated from (X^) and ((/),)„ 

386. The system of equations (203 31 ) is, by itself, a particular 
system of integral equations of the given differential equations, 
which does not contain any arbitrary constant. For the derivative of 
either of them, involving no arbitrary constant, must be satisfied by 
means of the equations (199 12 ) and (203 31 ), without any aid from the 
equations (203 24 ). The derivative of each of the equations (203 24 ) 
is, for the same reason, satisfied by the same equations (199 12 ) and 
(203 31 ), without the assistance of the equations (203 24 ). 

387. The functions (f) may be supposed to be introduced as 
the variables instead of the given variables (% { ). By this substitu- 
tion, the proposed system of differential equations assumes the form 



— 205 — 
Dx = X, 

By this same substitution in the equations (203 24 ) and (203 31 ), the 
equations (203 31 ) may be readily reduced by processes of elimination 
to an equal number of equations of the form 

fi = F i} 

in which the functions (F f ) do not involve those of the functions (/,) 
of which the values constitute the first members of these equations. 
Hence the derivatives of these equations, reduced to a finite form 
by the substitution of (205 2 ) become of the form 

D(F i —f i ) = XD x F i =0, 
or 

D x F ; = 0. 

But this equation does not involve either of the functions (/,) 
which are not contained in (F { ), and, therefore, cannot depend upon 
the equations (205 8 ). It is, therefore, identical, the functions 
(F;) are independent of x, and the equations (203 31 ) from which they 
are derived, contain only the functions (f). The substitution in 
(203 31 ) of the arbitrary constants (a ( ) for the functions (f) to which 
they are equivalent, reduces these equations to conditional equa- 
tions between the arbitrary constants. These equations (203 31 ), there- 
fore, represent the conditional equations, to which the arbitrary constants of 
the integrals of (199 12 ) must be subject, in order that they may coincide with 
the particular system of integral equations, to which the equations (203 31 ) 
belong. After the introduction of the functions (f), instead of the 
variables (.r ( ), into the functions (cp^, these functions (cp { ) can, by the 



— 206 — 

substitution of (205 8 ), be freed from all the functions (/,) which are 
not contained in (i^ : ). The derivatives of (c^) when thus reduced 
become, by means of the equations (205 2 ), of the form 

I)cp i = XI) x cp i =0, 

which must vanish independently of the equations (205 8 ), and, 
therefore, the functions ((p { ) do not involve x. Hence, by the substitution 
°f (%i) f or {fi) th e equations (203 24 ) give the values of (/^) in terms of (a t ). 
388. From any one given integral equation, denoted by 



w = 



the whole system of integral equations, to which it belongs, can be 
readily obtained. For the finite equation, to which the derivative 
of this equation is reduced by the substitution of the given differen- 
tial equations, is, from the very nature of the problem, another of 
the required system of integral equations. The derivative of this 
new equation gives a third integral equation, and the continuation 
of this process leads to the final determination of the whole of the required 
system of integral equations. 

389. This process of deriving a system of integral equations 
from one of its component equations, affords the means of testing a 
proposed equation, and ascertaining whether it be an integral equa- 
tion. For as great a number of independent integral equations is 
not admissible as that of the variables themselves ; if, therefore, the 
application of the process to a proposed equation conducts to a number of 
independent equations equal to that of the variables, it is a sufficient proof 
that the proposed equation is not an integral equation. 

390. When a system of integral equations contains superfluous 
arbitrary constants, that is, constants, which remain in the functions 
((pi), after the system is reduced to the form given in § 383 ; such 



— 207 — 

constants supply the means of obtaining other integral equations 
which are not contained in the given system. Thus if (206 n ) denotes 
an integral equation, from which the proposed system may be sup- 
posed to be derived, so that, reciprocally, this equation may be 
derived from the proposed system, and, therefore, 

in which F is any arbitrary function ; and if the notation is adopted 

^=\T,(jvD ft ii), 
in which arbitrary constants are denoted by (/,) ; the equation 

is also an integral equation. For the equation 

Du = 
gives, by direct differentiation, 

But it is obvious, from the form of (207 6 ), that the derivatives 
of u with reference to those of the constants (/?.;), which are elimi- 
nated from the functions (9);) and to which these functions are equal, 
are, themselves, functions of (cp { — /5 £ ) and 1/^ ; whereas the deriva- 
tives of u with reference to the superfluous independent constants 
(/?,-), which are contained in the functions ((jp £ ), are not merely func- 
tions of (cfi — fa) and i|v Hence the integral equation (207i 3 ) is a 
new equation, if it contains the derivative of u with reference to 
either of the superfluous constants (/?,-), and there are as many of 
these new equations as there are superfluous constants. But the 
number of independent integral equations thus obtained, is, of course, 



— 208 — 

subject to the condition, that it cannot exceed the number (n) of 
the independent solutions of the equation (200 2 ). 

391. Of all systems of integral equations, that, in which the 
arbitrary constants are the values which the variables themselves 
assume for a given value of one of them, deserves especial consider- 
ation. To simplify the discussion of this case, and place it in the 
position, in which it will best illustrate the problems of mechanics, 
the variable {x), of which the value is given, may denote the time, 
and the given time is the epoch or origin, at which the elements of 
the system of variables are given, and from which the variations are 
estimated. The values of the variables at this beginning of time 
may be termed their initial values, while those at any subsequent 
time are their final values. The differential equations express the 
laws of change, under which the variables pass from their initial to 
their final values, and are equally compatible with any proposed 
combination of initial values. The initial values are, therefore, ivholly 
arbitrary and independent. Their number is equal to that of the variables 
(%i), and, consequently, equal to the ivhole number of independent arbitrary 
constants, which is required for the complete integral equations. 

The epoch is also arbitrary, and seems to introduce an addi- 
tional arbitrary constant. But this constant is obviously superflu- 
ous ; it corresponds to the arbitrary position of the problem in 
time, without involving any modification of the essential conditions ; 
and is the complement of the arbitrary element, which is not 
expressed, and in reference to which the derivatives in the equations 
(199 12 ) are supposed to be taken. 

392. The passage, down the stream of time, from the initial to 
the final values, conformably to the conditions of change expressed 
in the differential equations, may be imagined to be reversed and, 
in a retrograde transit, the same laws of change would, by their 
reverted action, restore the variables to their initial values. In the 



— 209 — 

direct action, the initial values constitute the cause, and the final 
values are the effect ; whereas, in the reverted action, the final 
values become the cause of which the initial values are the effect. 
Hence it follows that, in any integral equation between the final and the 
initial values of the variables, the final and initial values of each variable 
may be mutually interchanged, and the resulting equation, if not identical with 
the given equation, is a new integral equation. In making this change, the 
sign of the variable, ivhich expresses the interval of time, must be reversed, 
because the interval, which is positive with reference to the initial 
epoch, is negative with reference to the final epoch. If, indeed, the 
interval were expressed, by means of the initial value (x ) and the 
final value (x) of the time, in the form (x — x ), its sign is directly 
reversed by the mutual interchange of the initial and final values, 
which transforms its expression to (x — x). 

393. Let F° denote the form, which any function F of the 
final and initial values of the variables assumes after the mutual 
interchange of these values ; and let 

#° = fjpi, 

represent the system of integral equations reduced so that the 
functions ((/),) do not involve the initial values (%{). The inter- 
change of the initial and final values in this system, produces a 
system of integral equations in which each variable is expressed in 
terms of that one variable, which represents the time, and of the 
arbitrary constants which are the initial values of the variables. 
This new system is represented by 

394. The discussion has, hitherto, been limited to differential 
equations of the first order, but it can, readily, be extended so as to 

27 



— 210 — 

embrace those of higher orders. If, for instance, the equations are 
given in the form 

in which the functions (X { ) may involve all the derivatives of the 
variables (x t ), which are of an order inferior to (p?), each of these 
inferior derivatives may be regarded as an independent variable, 
expressed by the form 

x^ = D]x i . 

With this new system of variables the given equations are 
replaced by the differential equations of the first order, represented 

by 

Dx { ?- 1) = z { ?\ 
Dxp~^ = X i , 
Dt=l. 

The number of these differential equations of the first order is 
easily seen to be (-2^ -f- 1) . 

395. When the differential equations are not given in the 
normal form (210 3 ), they can always be reduced to this form. For 
this purpose, each of the equations, which contains none of the 
highest derivatives of the variables, must be differentiated as many 
times, denoted by a,, as are necessary to raise it to an order, which 
contains such derivatives. If the given equations are represented by 

the equations, which are thus derived from them, may be expressed 

by 



— 211 — 

in which a • is zero, when it is applied to an equation which is not 
differentiated. Each of the derived equations contains at least one 
of the highest derivatives of the variables, which may be expressed 
by Z^ i+a iXi. The functions ((/),) should be independent functions of 
these derivatives ; whenever this is not the case, such derivatives 
can be eliminated from the derived equations, and one or more 
resulting equations will be obtained in which they are not involved. 
The independence of the functions ((/>,-) can, however, be directly 
tested by means of their determinant (185 29 ), which vanishes when 
it is taken with respect to quantities, for which these functions are 
not independent. 

When the functions ((/>,) are independent with respect to the highest 
derivatives contcdned in them, the required normal equations (210 3 ) are 
obtained from the given equations and their successive derivatives of an order 
not higher than those of the derived equations (210 31 ) by the usual process 
of elimination. For, 

First, there is a sufficient number of equations, because the 
number of equations, added to the given equations by differentiation, 
is 2^ which is the same with the number of derivatives, superior 
to the order (p { ), the highest of which are to be retained in the 
normal equations. 

/Secondly, these equations are independent of each other in 
respect to the derivatives of the order (p { ), and of the superior orders, 
and, therefore, sufficient for the required elimination; because if any 
of the equations of the inferior orders were not independent, their 
derivatives, which are included in the group, (210 31 ) would not be 
independent of each other. 

396. When the functions (y,-) are not independent with 
respect to the highest derivatives contained in them, each of the 
equations of an inferior order, obtained from- the derived equations 
by elimination, can be substituted for one of the derived equations, 



— 212 — 

which is necessarily involved in the elimination by which the 
reduced equation is obtained. If, therefore, one of the given 
equations is involved in the elimination, the order of the given 
equations is reduced by the substitution of the given equation. But 
if all the equations, necessarily involved in the elimination, were 
derived by differentiation from the given equations ; and if a 
denotes the smallest number of successive differentiations, by which 
either of these derived equations was obtained ; the reduced equa- 
tion is obviously a derivative of the order («) of an equation, 
which can be obtained by direct elimination from those of the 
given equations, which are of an order inferior by (a) to the 
derived equations, combined with the derivatives of the other 
given equations of an inferior order. This reduced equation of 
an inferior order may, then, be substituted for either of the given 
equations of a higher order, upon which its elimination neces- 
sarily depends. In all cases, therefore, in ivhich the functions (<jp,-) are 
not independent ivith respect to the highest derivatives contained in them, 
the order of the given equations can he reduced by the substihdion of 
an cqucdion of an inferior order obtained by elimination between some of 
the given equations and the derivatives of others, which are of an inferior 
order. 

397. That the normal forms, obtained by the process of § 395, 
are, as it was remarked by Jacobi, those which are obtained with the 
least complexity of operation, is easily perceived without any 
attempt at demonstration. It is, also, obvious, by what modes of 
substitution other normal forms can be derived from these, which 
are equivalent to them in the aggregate order of differentiation, but 
differ in the distribution of the derivatives. Thus if either of the 
functions (Xi) is of an order inferior by (qi) to that of the given 
equations, it is by (q { ) successive differentiations elevated to an order 
which contains one or more of the highest derivatives involved 



— 213 — 

in the normal forms. The ($-,-) th derivative of the equation (210 3 ), 
after the values of the highest derivatives, given by the normal 
equations, are substituted in its second member, so that it is 
expressed in the form 



D ^ Xi = X\ 



" 5 



may take the place of this equation in the system of normal equa- 
tions. If then D r /~ q iX i ' is one of the derivatives contained in (X { ), 
and if the normal equation (210 a ) is reduced to the form 

it may take the place of the equation 

■U t' %i' —— Xi/ 

in the group of normal equations. By means of (213 14 ) and its 
derivatives of an order inferior to the (<7;)th, all the other equations 
may be reduced so as only to contain derivatives of (%f) of an order 
inferior to the (p v — <7,)th. The normal system is by this means trans- 
formed to another normal system, in ivhich the highest derivative of one of the 
variables is increased, just as much as that of another of the variables is 
decreased. 

398. The repetition of the process of the preceding section 
may be so conducted that one or more of the variables shall finally 
disappear from the system of normal equations, and the number of 
equations will be simultaneously diminished to the same amount as 
that of the variables. The process may be continued, indeed, until 
only two variables remain, one of which is the variable (t), with 
respect to which the derivatives are taken ; but the reduction to 
this form involves the greatest prolixity and complexity of computa- 
tion. There are special cases, however, and particularly that of 



— 214 — 

linear differential equations, in which this mode of reduction is 
peculiarly advantageous. 

The principal portion of this discussion of differential equations 
is the combined result of the investigations of Euler, Lagrange, 
Cauchy, and Jacobi ; but an important addition to these researches 
is now to be developed, for which geometry is eminently indebted 
to Jacobi. 



THE JACOBIAN MULTIPLIER OF DIFFERENTIAL EQUATIONS. 

399. The function, which was called by Jacobi the neio multiplier, 
in order to distinguish it from the Eulerian multiplier, but which, on 
account of its superior importance, is here distinguished simply as 
the multiplier of a linear partial differential equation of the first order 
represented by (200 2 ), is that function which, multiplied by this equation, 
renders its first member an exact functional determinant (^„) of the indefi- 
nite function (/) and of n undefined functions (f) with respect to the (n -f- 1) 
variables (x ( ), zvhich are the independent variables of the given equation. On 
account of the mutual relations of the partial differential equation 
(200 2 ) and the simultaneous differential equations (199 12 ), this same 
function may also be regarded as a multiplier of the differential equations 
(199 12 ) ; and, for the same reason, it may be considered as a multiplier 
of the linear partial differential equation of the first order (200 20 ) of n 
independent variables. 

400. If either of the functions (f), or any function of these 
functions, is substituted for /, the determinant vanishes, by § 352, 
and the equation (200 2 ) is satisfied. The functions (f) are, therefore, 
n independent solutions of the equation (200 2 ). 

401. If the multiplier of the equation (200 2 ) is denoted by 
^®Mt>, the condition, by which the multiplier is defined, is expressed by 



— 215 — 
the identical equation 

The equality of the coefficients of Zk./in the two members of 
this identity is, by the notation adopted in the theory of determi- 
nants, expressed by the formula 

The substitution of this value of o/b^in the equation (189 2 ) gives 
the equation 

2 i D Xi (^M s >X i ) = 0, 

"which is a linear partial differential equation of the first order, by ivhich 
the multiplier is analytically defined. 

402. The defining equation of the multiplier may by (199 12 ) 
be developed into the form 

^(XiD^.^Ms -f yJztioD Xt Xi) = S^D^ys^fLD^ + ^A.I ; ) = 0, 

or 

f n ^Ms -f idA> ZiDz.Xi = D ^((d _j_ <sdi> S^Xt = . 

This equation divided by <J$Ms becomes 

F n log ob4 -f SiD^Xi = D log oaA _|_ S^X, = . 

If all the variables are regarded as functions of x, and if x is 
introduced in place of the element of variation, by means of the 
formula 

Dx = X, 



— 216 — 

the preceding equation finally assumes the form 

XD X log ^Mo + ZiD^Xi = ; 

which is an equation involving common differentials, by which the multiplier 
is analytically defined. 

403. The equation (215 8 ) gives, by (194 8 ), when 

t = 0, 

the value of the multiplier in the form 



JUMd 



X ' 



404. If the values of (/,) are expressed in terms of (#,-), by 
means of the equations (189 ]2 ), and if, by reason of the integrals 
(199 10 ), the constants (a,) are substituted for (/;), the value of the 
multiplier becomes 

^ m — V ) X'^Da^D^F, £ an F n > 

in which the sign may be rejected at pleasure. 

405. In the particular case, in which the equations (189 ]2 ) 
assume the form 

%i = (pi, 

in which the functions (g^) involve the arbitrary constants (a,-), 
together with no other variable than x, the value of the multiplier 
is by (189 12 ) reduced to 



<£}Md 



XS±D <h ^ Az 2 ^ F>a n % ~~ X2+. Da^Da.^ D a ,x n 

1 1 

XS±D A x y D h x i D fn x n — Xqj^'> 



— 217 — 

which equation might have been directly deduced from (216 n ) and 
(187 10 ). 

406. If the functions (F { ) are given independent functions of 
(/,), they are independent solutions of the equation (200 2 ) and give 
a multiplier (uatfc,-) different from >J2ii, and which is determined by 
the equation derived from (216 n ), 

X^ = S± D Xl F x D Xi Fz D Xn F n . 

This equation, by means of (186 14 ) and (216 n ), assumes the form 

X^Mo i =% 1}a 2±I) fl F 1 I) f2 F 2 D fn F n 

= X^M>Z±D fl F 1 I) fi F 2 D fa F n , 



which gives 



^= Z±D A F x Df 2 F 2 D fa F n . 



The second member of this equation is a function of the func- 
tions {fi), and may be an arbitrary function of these functions, so 
that it can have n independent values. The equation, therefore, 
serves to determine n -J- 1 independent values of the multiplier 
(vjb^d,.), which is, by (215 12 ), the whole number of independent values 
of which it is susceptible. Hence, the ratio of any two multipliers is a 
solution of the equation (200 2 ). It also follows from this argument that 
every solution of the equation (215 12 ) is a value of the multiplier. 

407. In the particular case, in which 

2,1).^ = 0, 

one of the n -\- 1 solutions of (215 12 ) is reduced to a constant, so that 
in this case, the constant must, contrary to the ordinary usage, be 
included among the solutions of the equation. The constant may 

28 



— 218 — 

be supposed to be unity, and, therefore, one of the multipliers of the equa- 
tion (200 2 ) is unity, ivhen the condition (217 2V ) ^ fulfilled, and all the other 
multipliers are solutions of the equation (200 2 ). 

408. When the solutions (/ { ) of the equation (200 2 ) are known, 
the corresponding value of the multiplier may be determined from 
(216 n ). But it can be derived by a shorter process, when either 
of the solutions (usM^.) of (215 12 ) is known, and also the initial value 
of oh>. Thus if II denotes the ratio of ua^ to <snMo, the equation 
(216„) gives by (194 7 ), 

When the initial values are substituted in this equation with 
the notation of § 393, it becomes 



qAd° ' 

The value of IT may, by the elimination of the variables (xf) 
be reduced to a function of the functions (//) ; and, if in this 
expression the functions (f) are substituted for their initial values 
(/!), the value of H is reproduced. For the function, which is 
obtained by this substitution, is a function of (/*) and therefore a 
solution of the equation (200 2 ) ; and it is, moreover, that particular 
solution, of which the initial value is the given function JI°. 

409. In the especial case, in which the initial values of (f) are 
the variables (x { ), the value of ok>° is obviously reduced to unity and 
the equation (218 u ) becomes 

410. When, in the differential equations (199^), the arbitrary 



— 219 — 

element of variation is assumed to be the variable x, the value of X 
is unity ; and, in this case, the equation (218 n ) becomes 

IT" == ^^ 



which in the case of the preceding section is reduced to 

and when, moreover, the equation (217 27 ) is satisfied, so that one of 
the multipliers is unity, this value is still further reduced to 

n° = 1. 

411. The arbitrary constants («,-) may be substituted for the 
functions (/,) in the equation (218 n ), when it is regarded as result- 
ing from the integrals of (199 12 ). By this substitution IT becomes a 
function of the arbitrary constants, which may be represented by C, 
and the equation gives, by means of (187i ), 

® hn = Z± D ai X x Da 2 X 2 DaX n = 



<£}MsiX' 



The logarithm of this equation becomes by the substitution of 
(216 2 ), and including C in the constants of integration, 

\0gZ±Da 1 X 1 I) a2 X 2 Da n X n = log^- + log C— log vflA, 

in which all the functions (X { ) can evidently be multiplied by any 
common factor, without disturbing the equality. 
412. In the especial case of 

X=l 



— 220 — 
the preceding formula becomes 

log 2 ± Da^Da, Da n X n =J2 i D x X i . 

413. When simultaneous differential equations are transformed 
from one system of variables to another, the multiplier usually under- 
goes a change at the same time, but there are conditions, to which 
the arbitrary element of differentiation may be subjected, and under 
which the multiplier remains unchanged. Thus if the new system 
of variables is represented by (?<>;), if the equations (199 12 ), in their 
new form, are represented by 

in which the accented sign of differentiation refers to the new 
arbitrary element of differentiation, and if 

u — a 

15 — °~> 

the values of ( W t ) become, by (199 2S ) and the preceding formulae of 
this section, 

Wi = GDiVi = G2 k (Dx k iv { Dx k ) 
= G2 k {X k D Xk tv i ) = GF n iv i . 

This value of ( Wi), in combination with the formulae (199 28 ) and 
(215 2 ), gives 

2 { ( W t D Wi f) = G2 k [X k 2 { (D Wi fD Xk w t )-\ 
= G2 k (X k D x J) 



— 221 — 

If oN is a multiplier of (220x2), the defining equation of (215 2 ) 
is, in respect to this multiplier, 

oN-^,.( WtDvj) = Z± D w fD Wl f x D w J n . 

The ratio of the equations (220 27 ) and (221 3 ), reduced by means 
of(186 13 ) and (187io), gives 

G df_ _ Z+DJD^f , Z>„„/„ 

-*- I U W X ±Jw^X-y JJw n % n 

= {2 ± D x wD Xl w x D,w n )-\ 

If, therefore, the multipliers gN* and ^(t are equal, the value of 
G becomes G\ if 



} 



G' = JS" + D w xD Wl x x D Wn x n 

= (JS" + D x tvDx x iv x Dx n iv n )~ x . 

414. The equation (215 25 ), applied to the new system of varia- 
bles (tOf), gives, by means of this equation and (220 17 ), if the multi- 
pliers are, for the instant, assumed to be equal, 

S t D w . W i = — D' log «jafl> = —G'D log <sM> 
= G'ZiD^Xi. 

415. If the arbitrary element of differentiation is supposed to 
be the same in both systems of variables, the values of G, W t , and 
cN* become 



G—l, 



— 222 — 

416. If the first m -j- 1, only, of the variables (x { ) are 
exchanged for the new variables («#,), which limitation is expressed 
by the formula 

the value of G' is abbreviated to 

G' = S + D w xD Wl x x D Wm x m 

= {Z±D x ivD XlWl D Xm w m )-\ 

417. Hence if the arbitrary element of differentiation, com- 
mon to the two systems, is one of the variables and is expressed by 
t, so that the remaining variables are still denoted by (x { ) and (w,), 
the formula (221 15 ) continues to express the value of G' . 

418. If the last {n — m) of the variables (tVi) are solutions of 
the equation (200 2 ), the corresponding values of the functions ( W { ) 
vanish by (22O22). If the multiplier is also supposed to remain 
unchanged, the partial differential equation (200 2 ), by which it is 
determined, is reduced to 

m 



The arbitrary constants (/?,-) may, therefore, be substituted for 
the solutions (^), and the value of G' becomes 

G = 2 -T D w XJJw 1 X 1 ■^ w m X m-^P m +i X m + l-^'P m +2 X m + 2 -L'PnZ'n' 

419. But if, instead of the equality of multipliers, the ele- 
ments of differentiation are identical in the systems, the defining 
equation is expressed in the slightly different form of 

Z i D lVi ( ; G'<sdkW i ) = 0, 



— 223 — 

in which the functions ( W t ) and the multiplier (oN*) are given by 
(22V). 

420. If the variables (w> € ) which are retained, coincide with 
the original variables (x { ), the equation for the multiplier becomes 

m 

o 
in which 

G = 2± I>p m+1 z, n+1 Dp m+2 x m+ 2 Dp n x n 

= {2 ± Dx m+1 w m+1 D Xm+2 w m+2 D Xn w n )- x . 

By the formulae of this and the two preceding sections the 
multiplier of the system of differential equations, to which a given 
system is reduced by means of any of its integrals, can be obtained 
from the multiplier of the given system. This will, soon, appear to 
be one of the most important properties of multipliers. 

421. If the given differential equations are of an order, which 
is higher than the first order, and have the normal form (210 3 ), the 
equation (215 25 ), by which the multiplier is defined, is simplified by 
the consideration that 

i 

The multiplier of the given equations, or of the equations (210 15 ), by 
which they should be replaced, is, therefore, determined by the equation 

2) log ^tt> -f- Ji^-nJ; = 0. 

422. If the functions X { do not involve #/ p ; -1) or if, in general, 

2 i Djp-DX i = 0, 

unity is one of the values of the multiplier of the given equations. 



_ 224 — 

423. If the given equations have not the formal form, but 
have the form 

such that they involve no derivatives of a higher order than the nor- 
mal forms, to which they are reducible by immediate elimination 
"without differentiation, the equation for determining the multiplier 
assumes a simple symbolic form, by means of the notation 

For it is to be observed that each of the subsidiary terms, of 
which the second term of the equation (223 25 ) is the aggregate, is to 
be obtained from the equations (224 3 ), by taking their derivatives 
relatively to x ( p~ 1] on the hypothesis that a$i) are functions of this 
variable, and thence determining, by elimination, the values of these 
subsidiary terms.. Hence if 



ffi — DsPi-Vatei) 



the derivatives of (224 3 ), relatively to x ( /'i~ 1) are represented by 
(177 24 ), provided the letters t of that equation are accented * times, 
and the number k is written below the u. From the comparison of 
(180 18 ) with (224 n ), it appears that (i,k) vanishes in the present case, 
and that the sign of d is to be reversed, whence the equation (180 26 ) 
becomes 

2S b tW = — Jlogflt., 

The equation (223 25 ) by which the multiplier is determined, assumes the 
symbolical form 

D log <shMd = — S h W = d log $>„. 



— 225 — 

424. It may, sometimes, happen that the values of a k i] and 
da$ are such that the sum of da^, and of XDa { k ] , in which X is 
constant, is simpler than da 1 ,!' 1 . In this case, if 

d' = d-{-lD, 

the addition of 

Z>log«* = X2>log«., 

to the equation (224 31 ) gives the symbolical form 

D log (ua4 sjj,J) = <T log 9L B . 

425. If the given differential equations have the form (210 27 ), 
so that they cannot be reduced to the normal form without differ- 
entiation, the equations (210 31 ), which are derived from them by 
differentiation, give, by direct elimination, a system of normal forms, 
which include, as a reduced system, the normal forms finally obtained 
by the process of §395. The multiplier of the equations (210 27 ) is 
determined by the symbolic equation (224 31 ), or (225 10 ), provided 
that in the values (224 10 ) of a{ i] and daf from which <) l( 3&„ is consti- 
tuted, the value of^. is increased by a k . 

426. The values of aft and daf may be determined directly 
from the equations (210 27 ). For this purpose, if X is written instead 
of a in order to avoid the confusion which might arise from the use 
of a as an arbitrary constant, and if the ingenious notation, which 
is familiar to the German mathematicians, for the continued product 
of all the integers from 1 to X inclusive, 

Xl = X(X — l)(X — 2) 3.2.1, 

is adopted, the equations (210 31 ) are represented by 

29 



— 226 — 
and we find, by well-known formulas, 

D x u)(p = k v D} [D x m FD x Mx M '] 

v' 

The inferior limit v is determined by the condition that neither 
V nor I — % -4- v' can be negative. Hence 

if Ji_|_l> x , v'=0, 

if I — l<>r, v' = x — X. 

In the former of these two cases the last term is 

! — D X ~ K D F- 



but in the latter case it is simply 

It follows, then, from (224 ]0 ) that, since F { does not contain any 
higher derivative of x k than p k , 

aV = D£iF t , 

427. The system of normal equations, derived by the process 
of § 395, is related to the system of normal forms, which has been 
discussed in the preceding sections, precisely as any reduced system 
of differential equations is related to that from which it is reduced 
by means of a portion of its integral equations. The integral equa- 
tions are, in this case, the equations (210 27 ) and all their derivatives, 



— 227 — 

which are inferior to the final derivatives expressed by equa- 
tions (210 31 ), the multiplier of the reduced equations is, conse- 
quently, obtained by dividing the multiplier <^Ms of the equations 
(210 31 ) by the function G given by the expression (223 9 ). The 
functions (to), involved in the value of G, represent the first mem- 
bers of the integral equations (210 27 ) and their derivatives. But it 
follows from (226 4 ) and (226 23 ) that 

D x iP L +»)D h t F i = D x iv K )F=ay. 

The equations (210 27 ) may, now, be supposed to be arranged in 
an order conformable to the orders of the derivatives, by which they 
are brought to the form (210 31 ), so that those, of which the higher 
orders of derivative are taken, may precede the equations of which 
lower orders are taken. Instead of reducing the equations, by a 
single step, to the final system, the reduction may be accomplished 
by successive steps ; and, at each step, the derivatives of the equa- 
tions (210 27 ), which are admitted into the group of integrals, may be 
diminished by unity, while the number of accents of the eliminated 
variables is also diminished by unity. At the step denoted by h, 
therefore, the derivatives of those equations (210 27 ) are added to the 
group of integrals for which the orders of derivative (A,) are greater 
than h. At this step a factor ( G h ) of G is also obtained, and all the 
derivatives of which it is composed are represented by the functions 
(«£>), in which the superior limit of Jc is the same with that of i. 
Hence if 

h — l i+1 >0, 

the value of the factor of G is 



— 228 — 
but if 

this factor is 

The logarithm of the complete value of G is, therefore, 

PRINCIPLE OF THE LAST MULTIPLIER. 

428. The consideration of the case in which there are two 
variables, leads to a valuable principle of integration, discovered by 
Jacobi, and which he called the principle of the last multiplier. In the 
case of two variables, the equation (215 2 ) becomes 

^ {XDJ+ X X D X J) == DJD X J X — D^fDJ,, 

which gives 

D x f x =^Jk>X 

B Xi f 1 = — ^MX 1 . 

Hence it is obvious that 

Df x = ^fc {XDx — X, Dx x ) 
or, by integration, 

f 1= r^(XDz — X 1 Dz 1 ), 

so that ivhen the multiplier is known, this equation determines the integral of 
the tivo differential equations (199 12 ) of two variables, or that of the sin- 



— 229 — 

glc equation to which they are equivalent, 

XD x z 1 — X 1 = 0, 

and the multiplier is, in this case, identical ivith the well-known Eulerian 
multiplier. 

429. When all the integrals but one of a given system of differential 
equations (199 12 ) are known, of ivhich the multiplier is also given, the last 
integral is determined by quadratures by the process of the preceding section ; 
because the multiplier of the two differential equations with two 
variables, to which the given system may, in this case, be reduced, is 
determined from the given multiplier by § 418. This is Jacobi's 
principle of the last multiplier. 

430. In the case of § 380, in which the element of variation (t) is one 
of the variables, if the functions ( JQ) do not involve (t), the equation (201 8 ) 
gives 



*=S;X 



X. 



from which / can be determined by quadratures, when all the other 
integrals of the given equations are known, even if the multiplier is 
not known, provided thatXj is reduced to a function of x i} by means 
of the known integrals. 

If the multiplier is also known, and if it does not involve t, the last of 
the integrals ivhich do not involve t can be determined by the process of the 
preceding section, and, therefore, the two last integrals of the given equations 
can, in this case, be determined by quadratures. 

Bid if the given multiplier (ue^Md) involves t, a multiplier (02X1(0^), which 
does not involve t, can be derived from all the integrals which do not involve 
t, and the quotient of these two multipliers gives by § 406, an integral involv- 
ing t, and which takes the place of (229 17 ) ; so that, in this case, the last 
integral is determined in a finite form without integration. 



— 230 — 

431. This proposition was shown by Jacobi to admit of the 
following generalization. If all the functions (X { ), in ivhich i is greater 
than m, are free from those of the variables (x t ) in which i is not greater 
than m, and if the remaining functions satisfy the equation 

in 


tivo integrations can alivays he performed by quadratures, whenever a multi- 
plier is known ivhich does not involve the variables {x i<m + 1 ), but ivhen the 
given multiplier does involve either of these variables one integration can be 
performed by quadratures, and another integral is given, immediately, tvith- 
out any process of integration. For if the given multiplier oaXfc involves 
only the variables (x i>m ), it not only satisfies the condition (215 8 ), 
but also on account of the equation (230 6 ) 

and is, therefore, a multiplier of the portion of the equations (199 12 ) 
in which i is greater than m. This portion of the given equations 
can, therefore, be first integrated, independently of the remainder 
of the system, and the last integral of this portion will be obtained 
by quadratures, because its multiplier is given. But the last inte- 
gral of the whole system may, also, be obtained by quadratures, 
because its multiplier is known; so that two of the integrals can be 
obtained by quadratures. 

But if the given multiplier involves any of the variables 
(x i<m+1 ), the separate integration of that portion of the equations 
(199 12 ) in which i is greater than m, gives a multiplier of this portion 
involving only the variables (x i>m ), which satisfies the equation (230 15 ); 
and by (230 6 ) it also satisfies the equation (215 8 ), so that it is a new 
multiplier of the given equation. The quotients of these two mill- 



— 231 — 

tipliers gives, by §406, an integral involving (.r !<H , +1 ),and which takes 
the place of the first of the two integrals, which are obtained by quad- 
ratures when the given multiplier involves only the variables [x i>m ). 



PARTIAL MULTIPLIERS. 

432. Additional to the systems of Eulerian and Jacobian mul- 
tipliers, and inclusive of them, are those, of which I have given the 
investigation in Gould's Astronomical Journal, and which I have called 
partial multipliers. The partial multipliers of the differential equations 

(199 12 ) are represented by (\sstk iik ^ ^J, in which i,l\,/i 2 , . . . etc. 

are any different numbers, or by (^eX(b /k ), in which 1 and K denote 
groups of numbers ; and they are defined by the equation 

P ua% = S ± D Xk f x D f t D f m 

in which P is any arbitrary function, Je 1: k 2 • • • % m are numbers not 
included in the groups I, and /i,/ 2 , etc. are solutions of the equa- 
tion (200 2 ). The notation (<J5vt(c (A) ) may also be used to denote the 
multiplier, with the definition that if 

iT denotes the group of numbers represented by (7e m ). 

433. The system of multipliers of (199 12 ), evidently, satisfies 
the system of differential equations, which are derived from (187 10 ), 
and represented by 

in which i includes all the numbers not belonging to the group I. 

434. The group of all the numbers not included in the group 



— 232 — 

(I) with the exception of any two, which may be selected at pleas- 
ure, may be denoted by II. The elimination of the corresponding 
values of X h from the equations, obtained from (200 2 ) by the substi- 
tution of the various values of (/;) gives the equations, which are 
represented by 

2 k (<mM s ( H ^X k ) = 0. 

This system of equations combined with that of (231 28 ) defines, 
analytically, the system of partial multipliers. 

435. In the formation of the multipliers, a careful regard must 
be had to their signs, conformably to the rule of formation of deter- 
minants, so that in general 

436. In the special case, in which the group (i, I) of § 433 
is reduced to a single number, and in which P is X, the preceding 
equations become 

XwMsi = — ok { , 
— X^fki -f- X.^ = 0, 
= J£ t D Xi (Xuafc,) = 2 { D Xt (^MoXJ ; 

so that, the multiplier is, in this case, the Jacobian multiplier. 

437. In the case, in which the groups (i, I) of § 433 include 
the numbers of all the variables but one, and in which P is unity, 
the equations become 

^k {i) = D Xi fi, 

so that, the system of multipliers is, in this case, that of the Eulerian multi- 
pliers amplified by Lagrange. 



— 233 — 

438. The partial multipliers may be denoted as the first, second, 
etc., to the last corresponding to the degree of the determinant which 
is the second member of the equation (216 n ). With this designa- 
tion, the last multiplier coincides with the Jacobian multiplier and 
gives a last integral of the differential equations, while the first mul- 
tipliers coincide with the Eulerian, of which each system gives a 
first integral of those equations. This proposition may be general- 
ized, and it may be shown that each system of multipliers determines 
an integral of the given equations hy means of quadratures, and holds a place 
in the rank of multipliers similar to that held hy the integral, in the rank of 
integrals. 

The investigation of the relations of the multipliers of differ- 
ent systems will be found to lead immediately to this proposition, 
after its truth has been established in the case of the Eulerian mul- 
tipliers. 

439. The deduction of an integral of a system of differential 
equations (199 12 ), by means of quadratures, from a given system of 
Eulerian multipliers, is quite a simple process. For the definition of 
these multipliers in § 437 gives 

If the quantities represented by ( #») are defined by the equa- 
tion 

Qi= C{^k (i) — D^ h Q k ), 

J x l 

i 

the required integral is 

f=2 i Q i = a: 

For the defining equation of Q t gives 
D x Z k Q k = ^>. 

1 

30 



234 — 



Hence it is found by differentiation that ( Q { ) is free from all 
the variables (x k<i ), for if this is supposed to be proved for ( Q k<i ) it 
it seen, by (232 27 ), that 



i — 1 h 



D X D X Qt = D {^k {i) — D x Z k Q k ) = D yaHP — D^D-Su Q k 

hi h v * 

= D Xk ^fe (i) — D x . ^k w = . 

The differential of (233 27 ) is, therefore, 

Df= S t {D x k k Q k D Xi ) =Z i (^Dx l ), 
o 

which corresponds to the required differential (233 19 ). 

440. When the differential equations (199 32 ) are transformed 
to other variables in the manner which is indicated in § 413, any 
multiplier of the new system is obtained by the following formula 
which corresponds to (231 15 ), 

P'$t H =2±D Wt AD f 2 D f m . 

If, then, the functions ( G) are defined by the equation 

G^ = 2±D vl x i D w x, D *, 

= (2± D x w h D x . iv h% D x . ^J -1 , 



the proposition (186 20 ) gives by (231 15 ) 

P'df H =I>Z I ( oa% £<*>) . 

441. If any of the solutions (/j-) of (200 2 ) are known, they 
can be assumed as new variables to take the place of either of the 
given variables, and the new multipliers must be determined by 



— 235 — 

the preceding equation. But it is evident that, in this case, the 
number of elements which compose each of the terms of (cNh) 
will be diminished by a number equal to that of the solutions, which 
are introduced as variables. Hence since m is the number of ele- 
ments which compose each term of ( *J2X(c 7 ), if (m — 1) is that of the 
known solutions, the number of elements of {g^ h ) ma y he reduced 
to one, in which case the multipliers (cN H ) become Eulerian and 
give the mth solution of (200 2 ) or the mth. integral of (199 12 ), by 
means of quadratures, which corresponds to the proposition of §438. 



III. 

INTEGRALS OF THE DIFFERENTIAL EQUATIONS OF MOTION. 

442. When the differential equations of motion are expressed 
in their utmost generality, there is no known integral which is suf- 
ficiently comprehensive to embrace them. But the equation (163 14 ) 
of living forces is an integral, which is applicable to all the great 
problems of physics, and holds the most important position in refer- 
ence to investigations into the phenomena of the material world. 
There are other integrals of great generality, which might be inves- 
tigated in this place, if the}^ were not clothed with such a character 
of speciality, that they properly belong to some of the following 
chapters. The application of Jacobi's principle of the last multi- 
plier to dynamic equations gives results of so general a character, 
that their investigation cannot appropriately be reserved for any 
chapter devoted to the consideration of special problems. 



236 



the application op jacobi s principle of the last multiplier to 
lagrange's canonical forms. 

443. It follows from the homogeneous nature of T (165 10 ), 
that each of Lagrange's equations (164 ]2 ), involves one or more of 
the quantities represented by (?/'), and the system of these equa- 
tions has, therefore, the form represented by (210 30 ). If, then, (a^°) 
denotes the coefficient of {rf k ) in the value of (w t ), given by (165 5 ), 
this value becomes 

o» i .= -2*(4 i) ^I), 
and that of T is by (165 u ) 

so that the functions (a[ l) ) only involve the quantities represented by 
(rj) and the time \t), and satisfy the equations 

4:4:4:. Each of Lagrange's equations may be expressed in the 
form 

g>, = D t 2 k {afrf k ) — i2 A>k (D v tf%ff h ) — Z>, i2 = 0. 

i i 

Hence, when £2 is only a function of (■)},) and t, the equa- 
tions (224 10 ) become 

k 
D^fi = <M f) = A«i° + Z h {D v afrf h — D v tf%) ; 



— 237 — 
from which are easily derived the equations 

k i 

k i 

The notation 

(i,Jc) = Z^Drfffc — D,<Wrf k ), 

k i 

gives 

ft*) = — (*>*)» 

In the substitution of these values in (224 31 ), it is evident from 
(180 18 ), (180 31 ), and (181 6 ) that the functions (i,k) disappear, and 
since D takes the place of D t , (224 31 ) becomes 

D log vje^Md = D log %> ni 

and, therefore, since the arbitrary constant may be neglected, 

which holds, even if the equations of condition involve the time. 

In all dynamical 'problems, therefore, in ivhich the forces arc indepen- 
dent of the velocities of the moving bodies, a Jacobian multiplier is given 
directly by the equation (237i 9 ), so that th & fast integral can alivays be 
obtained by quadratures. 

445. Hence, by § 430, in any dynamical problem, in ivhich the 
forces and equations of condition are independent of the time as ivell as of 
the velocities of the bodies, the two last integrals can be obtained by quad- 
ratures. 

446. The substitution of 

Ui^=x i \Jm i ,l Zi+x =y i \Jm i ,l 3i + i =iz i slm i , 






— 238 — 
in (164 20 ) and (162 28 ) gives 

2 2 

mvt = Ui + Z* i +i-\-Z' S i + 2 
Hence if 



2 



* 



the value of w^ is by (236 l5 ), 

which, combined with § § 346 and 348, gives 

sS>No = <& n = 2 M (W*)) 



in which ( 3 i ' l M) denotes the functional determinant of a group (M) of 
(11 -j- 1) of the functions (£;) relatively to the variables (17,). It may 
be observed that if % is the number of bodies of the system, and n 2 
the number of conditional equations, the value of n is 

n = 3 ??! — n 2 — 1 . 
447. If the conditional equations are represented by 

77=0, 
and if 

their derivatives with reference to (i; £ ) are represented by 

^(W.)==0. 
If then (JjT) denotes any group of n of the quantities (£,), and 



— 239 — 

(II,h) denotes a group of n-\- 1 of the same quantities in which the 
group (H) is included, the preceding equations give, by elimination, 
between all those in which i remains unchanged, 



"o v 



Since then the group (ff,h) is also denoted by (o&((d), if the 
group of all the remaining quantities (£ f ) is denoted by {N), if M' 
and N' are other groups of the same species, and if ( Q [N) ) denotes 
the determinant of the corresponding values of (c^), the preceding 
equations give, by elimination, 

which, it is easily seen, may be extended to the case of any groups 
whatever (M and M t ), in which each includes (w -J- 1) of the quan- 
tities {%{). If, therefore, some one group is arbitrarily selected and 
denoted by (M ), the equation (238 13 ) becomes 



-* = («S)^an i . 



448. If the derivatives of (rji) relatively to (5,-) are denoted by 

and if f €< l M) denotes the determinant of the values of (e^), which cor- 
respond to those of (bp) in ty ( ™\ the derivatives of (i; f ) may first 
be taken with respect to (£;), and if those of (§;) are afterwards 
taken with respect to (ij t ), they give by (186 20 ) 

1 = 2 ± D n riD n Vl D n i h D n Vn = 2i(Q<P<qP). 



— 240 — 

Hence, if gjY denotes the determinant of all the quantities (II { ) 
and (i]i) with reference to (£,-), the equation (239 13 ) gives 



JIH n J f)to(M) M \ n n J 



which, substituted in (239 20 ) reduces it to 



_ z N (& N) y 



(N: 



449. If there are no equations of condition, the value of \JhMd is 
reduced to 

= (S-±D 1l tD^ 1 D n l n f 

i ii 

= (2±'D^D^ ni D^ n y\ 

1 n 

If in this case, therefore, the values of (i] t ) coincide ivith those of (£;), 
the multiplier is reduced to unity. 

450. If the equations of motion were given in the system of 
§ 310, in which the forces, represented by the equations of condition, 
are included in those of 12, this system might, by means of the equa- 
tions of condition, be reduced to that of Lagrange's canonical forms. 
In performing this reduction, the equations of condition hold the 
same relation to the differential equations, which the equations (210 27 ) 
hold to the equations (210 31 ), in performing the reduction of § § 395 
and 425. It is also obvious that 

i i 

Hence the divisor by which the multiplier of the first of these 



— 241 — 

systems is reduced to that of the last, is by (228 8 ), (222 7 ), and the 
preceding sections 

{2 + DtfD^ D^ hl D< UP; IT, D^ n n f=^- 

1 n » + l » + 2 Sjij 2 

and, therefore, the multiplier of the system, previous to reduction, 

is by (240 8 ) 

451. If the system of differential equations is given in Ham- 
ilton's form, (166 3 ), the equation (215 25 ) for the determination of the 
multiplier becomes 

D log to* + S { (D n D a — D u D n ) II n , u = D log v*Jk = 0, 

i i i i 

whence the multiplier of this system is unity. 



CHAPTER XI. 

MOTION OF TRANSLATION. 

452. If the coordinates of the centre of gravity of a system 
axe x g ,y g , z g , and if those of any other point are x s -j- x i} y g -\-y iy 
z g -4- Si, the value of T becomes, by (162 28 ) and (164 20 ) and the con- 
ditions of the centre of gravity (155 19 ), 

T= i (x' g +yl+z' g )2 imt + ««[«*« +tf + 4)] 

31 



— 242 — 

Hence the motion of the centre of gravity is determined by 
the equation, derived from (164 12 ), 

Z imi D t af g = 2 i m i D i t x s = D x il, 

and the corresponding equations for the other axes. The value of 
£2 may be restricted in this equation to the external forces and those 
which correspond to the external equations of condition, for the 
internal forces and equations of condition being dependent solely 
upon the relative positions of the bodies of the system, are functions 
of the differences of the corresponding coordinates of the bodies, 
from which x g ,y g , z g disappear. 

The motion of the centre of gravity is, therefore, independent of the 
mutual connections of the parts of the system, and is the same as if all the 
forces ivere applied directly at this centre, provided they are unchanged in 
amount and direction. 

453. Since the second member of(242 3 ) expresses the whole 
amount of force r acting upon the system and resolved in the direc- 
tion of the axis of x, this equation expresses that the motion of the cen- 
tre of gravity in any direction depends upon the whole amount of external 
force acting in that direction. 

If, therefore, the ivhole amount of external force acting in any direc- 
tion vanishes, the velocity of the centre of gravity in that direction is uniform. 

MOTION OF A POINT. 

454. When the system is reduced to a single point, it becomes 
a mass united at its centre of gravity, and the only possible motion 
is that of translation. The position of the point is determined by 
three coordinates, which, combined with their derivatives and with 
the time, constitute a system of seven variables, and require, in gen- 
eral, six integrals for the complete determination of the motion of 



— 243 — 

the point. The differential equations become, in this case, if the 
mass of the body is assumed to be the unit of mass, 

D' 1 t x = D x n, 

with the corresponding equations for the other axes. 

A POINT MOVING UrON A FIXED LINE. 

455. The two equations by which the line is defined are two 
equations of condition, which may be denoted by 

Together with their derivatives, they take the place of four of 
the integrals of § 454. Of the two remaining integrals, ivhen £2 docs not 
involve the time, both can be determined by quadratures by § 445. 

One of these integrals is, indeed, the equation of living forces 
(163 13 ), which becomes in this case 

z, 2 =2(J2-f H) = (I) t s) 2 . 
The final integral is obtained from this integral by the equation 

t== Js \/(2Si- 



\J (2 Si + 2 H) 



= 1 



D n s 



\/(2Sl + 2II) 



456. It follows from (243 ia ) that the velocity of a body only 
depends upon its initial velocity and the value of the potential at 
each point of its path ; and this conclusion coincides with the propo- 
sition of § 58. In ivhatever path, therefore, a body moves from one point 
to another, the increase or decrease of the square of its velocity may be meas- 



— 244 — 

ared by that of the potential, when the equations of condition and the forces 
zvhich act upon the system are, like the fixed forces of nature, independent of 
the time and the velocity of the body. 

457. If there is any point upon the line, beyond which the 
decrease of the potential exceeds one half of the square of the 
initial velocity, the body cannot proceed beyond that point. If there 
is, in each direction from the initial position of the body upon the line, a lim- 
iting point of this description, the motion of the body is restricted to the inter- 
vening space. Since the body can only have the direction of its 
motion reversed at the limiting points where its velocity vanishes, it 
must oscillate back and forth upon the whole of the intervening 
portion of the line, according to the law expressed by the equation 
(243 23 ). 

It is evident from the inspection of the equation (24323), that 
the time which the body occupies in passing from any point (A) of 
the line to another point (B), must be the same with that which it 
occupied in the preceding oscillation in the reverse transit from the 
point ( B) to the 'point {A) ; and, therefore, the entire duration of oscil- 
lation must be invariable. 

458. If the line returns into itself, and if there is no point 
upon it for which the decrease of the potential is as great as the 
initial power of the body, the body will continue to move through the 
whole circuit of the line, and will always return to the same point with the 
same velocity, so that the period of the circuit tvill be constant. 

459. When the forces and the equations of condition involve 
the time, the multiplier becomes by (238 13 ) 

<mM s = 2 x (I) v xf 

and the last of the integrals, zvhich are required to solve the problem, can be 
obtained by quadratures. 



245 — 



TIIE MOTION OF A BODY UPON A LINE, WHEN THERE IS NO EXTERNAL FORCE. 

CENTRIFUGAL FORCE. 

460. When the line is fixed, and there is no external force, £2 van- 
ishes in (243 19 ), and the velocity is, therefore, constant. 

461. In this case, the line may be regarded as the locus of a 
resisting force, which acts perpendicularly to the line. The plane of 
x and y may be supposed to be, for each instant, that of the curva- 
ture of the line at the position of the body, R may be the resisting 
force of the line, and o its radius of curvature ; and elementary con- 
siderations, combined with the equation (164 25 ), give 



D t x = D t s sin p = v sin p 



X 1 



J)*x = vcos p z D tx = v 2 cos x I) sx = ^cos x =Bcos x , 

whence 

R=-, 

Q 

so that the pressure against the line is measured by the quotient of the 
square of the velocity divided by the radius of curvature, which is called the 
centrifugal force of the body. 

462. If there are external forces, the tvhole pressure upon the line 
is obtained by combining the action of all the external forces resolved perpen- 
dicularly to the line, with the centrifugal force. 

463. The centrifugal force cannot be used as a motive power 
in machinery, for the body moves perpendicularly to the direc- 
tion of this force ; and, therefore, the power communicated by it 
vanishes, because it is measured by the product of the intensity of 
the force multiplied by the space through which it acts. 

464. If the line is not fixed in position, but has a motion of 



— 246 — 

translation, the same motion of translation may be attributed to the 
axes of coordinates, so that the coordinates of the moving origin at 
any time may be a x , a y , a z , with reference to the fixed axes. If the 
coordinates of the body with reference to the moving axes are % x) % y , 
\ z , the value of 2 T (164 21 ) becomes 

2T=Z x (¥ x + a' x y 

= D t s 2 -\- 2 wD t s cos * -f- u?, 

2 
= s' -j- 2 IV §' COS * -j- 2V 2 

if iv denotes the velocity of the motion of the origin, and s the 
length of the line passed over by the body. Hence Lagrange's 
equation (164 12 ) gives 

D t (/ -J- iv cos * ) = D s (/ w cos £) = s'wD s cos £ . 

But, since the angles which s makes with the axes are inde- 
pendent of the time, the derivative is 

D t O cos i ) •= D t 2 X {w x cos • ) = 2 X (w x cos x -\- / w x D s cos %) 
= 2 x (w' x cos x )-\-s' D s (to cos ^), 

which reduces the preceding equation to 

D t s = — 2 X (w' x cos s x )= — Wcos £, 



if 



2 2 2 

W= \j(W x -\-W'y-\-W z ) 



denotes the acceleration of the line at each instant. Hence it is 
easy to see that if the acceleration is perpendicular to the line, the relative 
velocity of the body to the line is not changed ; but if the acceleration is in 
the direction of the line, the change of relative velocity is exactly equal to the 



— 247 — 

acceleration, so that there is, in this case, no change in the actual velocity of 
the bod// in space. 

465. It follows, from the preceding investigation, that if the 
motion of the line is uniform, the relative velocity of the body and the line 
remains constant. 

466. It is also apparent from this investigation that even under 
the action of external forces, the relative motion of the body to the line may 
be computed, by regarding the acceleration of the line as a force acting upon 
the body in a direction opposite to its actual direction. 

467. If the line rotates about a fixed axis, which is assumed to 
be the axis of z, let 

u be the projection of the radius vector upon the plane of xy, 
(p the angle which u makes with the rotating axis of x, and 
a the velocity of rotation, 

and the value of 2 T becomes 

2T=/-\- /+u 2 (<p'+a) 2 

= (B t s) 2 -\-2u 2 (p'a-\-u 2 a 2 

2 

= /-J- 2uas r cos£ -\-u 2 a 2 , 

in which 6 is the angle, which s makes with the elementary arc udcp. 
Hence the derivatives of T are 

D s T=s' -\- ua cos £ , 

D s T= / D s (ua cos 6) -(- « 2 m cos " ; 

and the equation (164 12 ) becomes 

D 2 s = — u cos & a -\- a 2 u cos ". 

The former of the two terms which compose the second mem- 



— 248 — 

ber of this equation, is the negative of the acceleration of the rota- 
tive velocity resolved in the direction of the arc of the rotating line. 
The latter term represents the centrifugal force, which corresponds 
at the body to the rotation («), and which is also resolved in the 
direction of the moving arc. But the centrifugal force is purely 
relative in its character, and arises from the resistance of the body 
to accompany the curve in its change of motion occasioned by rota- 
tion. These terms combined show, then, that in this case, as well as 
in that of translation, and, consequently, in every case the relative 
motion of the body to the line may he obtained by attributing to the body the 
negative of the acceleration of the line, which occurs at the position of the 
body ; in the case of external forces, their action must be united to that 
ivhich arises from the acceleration of the line. 

468. In the case of an uniform rotation about a fixed axis, the 
equation (247 29 ) becomes 

D 2 s — a 2 u cos " = a 2 uD,u. 

The integral of the product of this equation, multiplied by 
2D t s, is 

(D t sf=a 2 (u 2 -\-A), 
in which A is an arbitrary constant. Hence it is obvious that 

. _ r i _ r d u s 

469. When the constant (A) is negative, the value of u cannot 
be less than \j — A ; so that when the body approaches the axis, its 
velocity upon the line is constantly retarded, and vanishes, when its 
distance from the axis is reduced to \/ — A, after which the direction of 
the motion is reversed. If the portion of the line, upon which the 
body moves, extends at each extremity, so as to be at as small a dis- 



— 249 — 

tance as y/ — A from the axis, the body tvill oscillate upon it ivith a con- 
stant period of oscillation. 

470. When the constant (A) is positive, or when it is negative, 
and no portion of the line in the direction, towards which the body 
is moving, is at so small a distance as \J — A from the axis, the 
motion of the body upon the line will constantly retain the same 
direction. If, moreover, the curve returns into itself, the body will 
always continue to move around it, with a constant period of revolution. 

471. When the constant (A) vanishes, the equation (248 20 ) 
gives 



at 



D t s = au 

J s u J u it ■ 



If the curve, also, passes through the axis of rotation, the value 
of D u s may be supposed to be constant, while the body is very near 
the axis, and may be represented by /? ; so that the motion of the 
body in the vicinity of the axis is given by the equation 

a#= (i log u. 

The second member of this equation becomes infinite when u 
vanishes, and, therefore, the motion of the body, in this case, is infinitely 
sloio in the immediate vicinity of the axis. 

472. When the rotating line is straigJd, let 

p be the distance of its nearest approach to the axis of rotation, and 
£ the angle which it makes with the plane of x y. 

If then s is counted from the foot of the perpendicular, which 
joins the nearest points of the line and the axis of revolution, the 

32 



— 250 — 
value of u 2 is given by the equation 

U 2 =p 2 -\-S 2 CO$ 2 A ; 

whence (248 24 ) becomes, in this case, 

at ~ J \/ (p 2 +A+s> cos 2 d) 

= jko lQ g & cos * + v/(/+^+^ 2 cos 2 ^)] - log{ 2 P lte A) m > 

in which the arbitrary constant is determined so that t may vanish 
with s, and this equation is applicable when (p 2 -|- A) is positive. 
In this case, the substitution of the notation 

h % =f 4- A, 

tan re = -, 

reduces the preceding equation to 

a t cos & = log cot I <p . 
But when (p 2 -j- il) is negative, the substitution of the notation 

k 2 = -(p 2 + A), 

h 

sin if = s, 

T s cos 7 

and the determination of the arbitrary constant, so that t may vanish 
when s has its least possible value of Jc sec 6, reduce the equation 
(250 6 ) to 

a t cos 6 = log tan h if . 



— 251 — 

When (p 2 -J- A) vanishes, the equation (250^) is reduced to 

a i cos 6 = log — ; 



So 



in which s is the initial value of s. When p also vanishes, the sur- 
face described by the line is a right cone, and when it is developed 
into a plane, the path, described by the body, becomes a logarithmic spiral. 

473. When the rotating line is the circumference of a circle which 
is situated in the plane of rotation, let 

E denote the radius of the circle, 

a the distance of the centre of the circle from the origin, 

2 (p the angle, which the radius of the circle, drawn to the body, 

makes with that which is drawn in a direction opposite to 

the origin, 

and the equation (248 24 ) becomes 

, r 2E 

at = I 

Jib 



When A-\-(R — a) 2 is positive, which corresponds to the case of 
§ 470, let 



h 2 = A-{-(B-\-a) 2 , 



sin 2 i 



K xl -]- u> 2 
4aE 



V 



and by the notation of elliptic integrals of § 169, the equation (251 18 ) 
becomes 

at = -£■&&. 

When i is so small that its fourth power may be rejected, this 



— 252 — 
equation gives, by an easy reduction, 

at= (1 -f- i sin 2 /)— ~ — — sin 2 mn2<jp. 

In this case, therefore, the time of describing the semicircum- 
ference, for which 2 y is greater than a quadrant, exceeds the time 
of describing that for which 2 y is less than a quadrant by 

R . o. iafi 2 iaE 2 

T-snrV 



When J. -|- (it — a) 2 is negative, which corresponds to the case 
of§4G9, let 

• 2 - & 
sin- 1 



4afi : 



• . sin w 

sm 6 = —r-h 

sim 



and the equation (251 ]8 ) becomes 

2E C . 2Rsini C 

at = —r- J sec 6 = — -, — / sec 9 

When i is so small that its square may be rejected, the duration 
of an oscillation becomes 

a \ a 

When the circumference passes through the axis of rotation, a 
is equal to R, and the time of the small oscillation becomes identical 
with that of the semi-revolution of the circle ; but the time of a 
larger oscillation exceeds that of the semi-revolution. 



— 253 — 
When A-\- (R — a) 2 vanishes, the equation (251 ]8 ) becomes 

at = y-J secc J p = ^/-logtan(i n + ky). 

When A-\-(R — af is very small, and its ratio to iaR is 
denoted by d A, the equation (251 18 ) gives throughout the greater 
portion of the path, in which <p differs sensibly from i n, that is, in 
which the body is not near its point of closest approach to the axis 
of rotation, so that the square of dA may be neglected, 

= (1 — Id A) i/— log tan {in-\- £(p)- — id Ay/— tan 9 secy. 
But in the vicinity of the point of nearest approach, let 

XfJ = i 7T (p 

be so small that its square is of the same order with $.4, and the 
equation (251 18 ) gives 

at = — U- I = — i/-Sin [ ~ 1] -=, when d A is positive. 

= — 1/— Cos [ ~ 1] ,- • .. , when dA is negative. 

V a \f ( — 8 A) ' ° 



474. When the rotating line is ivholly contained upon the surface of a 
cylinder of revolution of which the axis is the axis of revolution, u is con- 
stant and the equation (247 29 ) becomes 

D (p sD 2 s = — u 2 a? , 

from which (p or s may be eliminated by the given equation of the 
curve. 



— 254 — 

475. When the velocity of rotation is constant, the second 
member of (253 28 ) vanishes, and the velocity of the body is conse- 
quently uniform. 

476. When the curve is a helix, the value ofZ>.s is constant, 
and the equation (253 29 ) gives 

M 2 

in which A is an arbitrary constant. 

477.' When the acceleration is uniform, a' is constant, and the 
integral of (253 28 ) gives 

(D t sf = u 2 a(A — <p), 

u 2 a't= I -jj-. ; 

J sS /(A—(p)> 

in which A is an arbitrary constant. 

MOTION OF A HEAVY BODY UI"ON A FIXED LINE. THE SIMPLE PENDULUM. 

478. When the line is fixed, and the force which acts upon 
the body is that of gravity at the surface of the earth, represented 
by g, and the axis of z is assumed to be the vertical, directed down- 
wards, the equations (243 19 _32) give 

v 2 =2gz-\-2H, 

t—c y r ^ s 

J s ^(2gz-\-2ff) —J zS /(2ffz-\-2Jiy 

479. If the curve is contained upon the surface of a cylinder 
of which the axis is vertical, the motion of the body is the same as 



— 255 — 

it would be upon the plane curve, obtained by the development of 
the cylinder into a vertical plane ; because the value of D z s is not 
changed by this development. 

480. If the fixed line is straight, the equation (254 2S ) becomes 

if ^' is the initial velocity of the body. 

481. If there is no initial velocity, the preceding equations 
become 

cos * gt = ^{2gs cos*) = v, 
or 

z gt g? 2 gs s ' 

482. If the curve is the circumference of a circle, the centre 
of the circle may be assumed as the origin of coordinates. If then 
the axis of z x is the intersection of the plane of the circle with 
the vertical plane, which is drawn perpendicular to it through the 
origin, and if R is the radius of the circle, and 

the equation (254 28 ) becomes 

2R 



r 2R. r 



, ? V / (23^cos^cos29 + 2#) J^sJ (2 H-j-2 g H cos* — 4 g E cos % sin 2 cp) 



If then H is greater than g Rcos^, which is similar to the case 
of § 470, let 

h 2 =2H+2gRcos z Zi , 
4^ R cos I = h 2 sin 2 / ; 



256 — 



and the preceding equation becomes 

4 2R Ct 



When i is quite small, this equation admits the same reduction 
with that given (251 31 — 252 3 ). 

If H is smaller than gR cos l 1} which is similar to the case of §469, 
let 

h 2 = 4(/Bcos z z sin 2 i 

. , sin m 
Sin 8 = -r— ■r- , 
sin i 

and the equation (255 24 ), becomes, by the same reduction with that 
given in (252 17 ), 



^vyy^, 



<9 

which when i is small gives for the time of oscillation of the simple pen- 
dulum in an oblique plane 

If H is just equal to^ijlcos*, the equation (255 24 ) becomes 

* = V / (7^) 1 °s tan ^ 7T + *»)■ 

The case in which H differs but little from gBcos^, may be 
subjected to the same treatment with that adopted in (255 5 _ 23 ). 



— 257 



MOTION OF A HEAVY BODY UPON A MOVING LINE. 

483. If the heavy body moves upon a line, which has a 
motion of translation in space, the equation of motion becomes, by 
the form of argument and notation adopted in § 464, 

]y\s = — J7cos* -}-# ,cos ^ 
484. If the motion of the line is uniformly accelerated and 
invariable in direction, the motion of the body upon the line is the 
same which it would be if the line were fixed, and the force a con- 
stant force which coincided in amount and direction with the re- 
sultant of g and — W. Thus if the line moves vertically downwards 
with an accelerated velocity, equal to that of a heavy falling body, 
the body moves upon the line with an uniform velocity. 

485. If the line is straight, and if the motion of translation 
follows a law, dependent exclusively upon the time, so that if 

A t denotes the law, by which the line moves in the direction 
of its length, the acceleration in the direction of the line is 

— Wcos^=D 2 t A r , 
and the value of s becomes 

s = a -j- It -J- a gft cos * -|- A t , 

in which a and b are arbitrary constants. The absolute motion of 
the point in any direction in space, as that of the axis oix x , is repre- 
sented by the equation 

x 1 =(s — A,) cos s Xi -\-p cos p x{ , 



— 258 — 

in which p denotes the perpendicular upon the line from the origin. 
If the line is vertical, and limited in its motion to the vertical plane 
of x x z u and if the axis of % is vertical, the equations which deter- 
mine the position of the point in space are 

g x = a -f- It -f- igt 2 . 

When p increases uniformly so that p is constant, these equa- 
tions give 

x x —p't, 

s 1 = a -J- —,x x -j- g— 7a x n 

so that the path of the body in space is a parabola, of which the 
axis is vertical. 

486. If the line moves with an uniform motion in a straight line, 
the equation (257 8 ) gives 

D 2 t s = g cos*. 
The integral of the product if this equation multiplied by 2 D t s is 

(I) t sy=f2gcoslD t s = 2( / f t I) t s = 2< / z-{-a, 

in which a is an arbitrary constant. Hence if 

V denotes the velocity of the translation of the line, 

the square of the velocity of the point in space is 

(n [Sl Y=W(2gz-\-a)-Vcos:Y + (Vsm:f 

= 2$z-\-a-\-V 2 — 2Vcos^ S /(2gz-i r a). 

The augmentation of the power of the moving body above its 



— 259 — 

initial power is, then, 

P=i(D tSl f— k{D t s°Y=g{z— s )— V{vco$ v s — v°co$ v s o). 

If the body had moved through the same path upon a fixed 
curve, the increase of power would have been 

Q=g{z — z o )-\-gVtcos r z . 

If P is greater than Q, the excess of P above Q is the power 
acquired by the body from the accelerating motion of the line. But 
if Q exceeds P, the excess of Q above P is the power communi- 
cated by the body to the line, which involves the theory of many 
machines, of which heavy bodies are the moving forces. If, for ex- 
ample, the line moves horizontally, the power communicated by the 
weight is 



l o' 



Q — P=zV(vcos*— y°cosTo). 

If, moreover, the initial velocity of the body, relatively to the 
line, vanishes, the expression of the communicated power is re- 
duced to 

e_P =Fc os^[2<K2-<)]; 

and when the direction of the line at its extremity coincides with 
that of its translation, this expression is still further reduced to 

487. If the line is the circumference of a vertical circle, of which 
the radius is R, and if (p is the angular distance of the body from 
the lowest point of the circumference, the equation of motion (257 8 ) 
becomes 

RB^y = — Wcosw — #sin (p. 



— 260 — 

When the motion of the line is in a vertical direction this 
equation becomes 

B&*(p = —( W+g) sin 9 ; 
which, when q> is very small, is reduced to 

RD*y = -{W-\-g)< 9 , 
The integral of this equation is 

( p = Asm(t S J^-{-b) i 
in which A and b may be determined by the equations 
2>J(gR)I) t \ogA=-Wsm2(t ) j!L+b), 

2^{gR)D t b = WA[l— ***(f\J £ + *)]; 

which give 

sl{gR)D t {Asmb) = — AW&m(tsJ ^-\-h)$m(tJ ^ 
= hA Tr[cos(2^t/|4- b) — cos b] 
=A W[sm 2 \b— sin 2 (if */| -f hb\ ; 

s/(gR)D t (Aco8b) = — AWsm(iJ ^-\-b)cos(t^^) 

= —iAw[s[n(2tJ^-j-b)-\-smb]; 

when TT is very small in comparison with #, J. and B may be as- 
sumed to be constant in the first integration of the second members 
of these equations. 

When W is dependent upon the position of the body in such 



— 261 — 
a way, that, if ^ is a function of time, 

the preceding equations give 

y/teB)Afab = —f t (Van(t } Jl)), 

If, for example, 

2T= 2hsmmt ; 



these integrals become 



• m 2 Ji—g 

^(2B)Aco S b =—^^sm(mt + t S /-£) — _A^gin(m* — <y/|) 

in which the arbitrary constants are determined so that J. and b 
vanish with the time. 

488. If the line rotates about the vertical axis of s, the equation 
of motion becomes, by the analysis and notation of § 467, 

J}fs = — UQOs&a' -j-« 2 wcos"-|-y cos * 
= — ucos&a -J- a 2 n D s u -4- g D,z. 

489. When the rotation about the vertical axis is uniform, this 
equation becomes 

D 2 s = a 2 uD s u -\-gD s z. 
The integral of the product of this equation multiplied by 



— 262 — 
2D t s is 

in which a is an arbitrary constant. 

490. When the rotating line is straight and passes at a distance 
p from the axis, if s is counted from the foot of the perpendicular 
(p) upon the line, the equation becomes 

(D t s) 2 = aV sin 2 * -]- 2gs cos£ -f" ft V 2 "f" a 

= (a s sin % -j- - cot'j -J- a -J- a 2 /? 2 — (-cot*j , 

of which the integral is easily found to be 

at sin * = log(a 2 ssin 2 *-}~ 2g cos*-|-2a sin $ z D t s) -\-b, 

in which b is an arbitrary constant. 

491. The integral, in this case, can be just as readily obtained 
from the equation (261 29 ) which becomes a linear differential equa- 
tion. Its direct integral is 

a cos' . a t sin I , n — atsvai 

s — \ • L = Ac z -\-Bc 

a sin | ' 

in which A and B are arbitrary constants. This form is identical 
with that given by Vieille in his solution of the particular case 
of this problem, in which p vanishes. 



492. If a<(^eotif—a 2 p 2 



the value of s must be such as to render the second member of 
(262 9 ) positive ; that is, the limiting values, between which the 
body cannot be contained, are defined by the equation 



a 



s sm s z =z — -cot* +i/|(-cot*j — ct 2 p 2 — a\ 



— 263 — 

The velocity of the body upon the line vanishes at these limits. 
If the initial direction of the motion of the body is towards these 
limits, it will approach them with a diminishing velocity ; and 
when it arrives at the nearest limit, the direction of motion will be 
reversed, and it will thenceforth continue to move away from the 
limits. 

If a = — a?p 2 

one of the limits is at the foot of the perpendicular (p), and the 
other limit is above this foot, at the point for which 

s=== -cot*. 

a z 

If #< — « 2 j0 2 , 

one of the limits is above the foot of the perpendicular, while the 
other is below it. But if 

a> — a 2 p 2 

while it satisfies the condition (262 25 ), both the limits are above 
the foot of the perpendicular. 



493. If «>(^cos^) 2 — «y, 



the motion will always continue in the same direction along the line, 
(a-\-a 2 p 2 ) will express the square of the velocity of the body upon 
the line when it is at the foot of the perpendicular. The point of 
least velocity upon the line will be determined by the equation 



g cos . 



and the least velocity will be 



— 264 — 

494. If a = (| cot if— ay 

the direction of the motion along the line is not subject to reversal 
for, in this case, the equation (262 9 ) becomes 

D t s = a s sin * -(- - cot* ; 
of which the integral is 

(OL v sin \ 
r + 1 ) • 
g cos * ' / 

The time of reaching the point, at which 

9. cos I 






a sin' 



2sJ 



that is, the point, at which the velocity vanishes, becomes infinite ; 
or in other words,, the body never reaches this point, at which its 
direction of motion is to be reversed ; or if the body is placed at this 
point without any initial velocity along the line, it will remain sta- 
tionary upon the line. 

495. If the rotating line is the circumference of a circle, of which 
the radius is R, let the origin be assumed so that the centre of the 
circle may be upon a level with the foot of the perpendicular (p), let 
fall from the origin upon the plane of the circle. Let then 

k denote the distance of the centre of the circle from the foot 
of the perpendicular, 

<p the angular distance upon the circumference of the body 
from the lowest point of the circumference, 

and the values of z and u, in equation (262 2 ), are given by the 



— 265 — 

equation 

2 = Rcosy sin? -|-^ cos?, 
u 2 = (k -j- R sin 9) 2 -f- (i> sin? — R cos 9 cos?) 2 

— p _|_ ^2 _|_ys gin 2 p _|_ 2 /<• R sin y _ j,, 7? s i n 2 ? cos 95 — R 2 sin 2 ? cos 2 9 , 

whence equation (262 2 ) becomes 

R 2 (D t (pf = a-\- a 2 (F -}- i? 2 -j-/ sin 2 *) + 2gp cos? + 2 a 2 /c R sin 9 
-j- 2 (y — « 2 jo cos?) R sin? sin 9) — a 2 R 2 sm 2p z cos 2 9 . 

The points of maximum and minimum velocity along the arc 
are, therefore, determined by the equation 

a 2 Jc R cos (fi — (g — a 2 p cos z ) R sin? sin tp x -f- a 2 R 2 sin 2 ? sin (p 1 cos 9)! = , 

and are, consequently, at the intersections of the circumference with 
the equilateral hyperbola, which is described in the plane and passes 
through the centre of the circle, of which one of the asymptotes is 
horizontal, and the polar coordinates (r 2 , 92) °f the centre, with 
reference to the centre of the circle, are given by the equations, 

r 2 sin 92 = — ^ cosec 2 ? , 

r 2 cos 9 2 = 4 cosec? — p cot ? . 



This hyperbola cannot cut the circumference in less than two 
points ; and there are four points of intersection when the distance 
from the centre of the circle to the nearest point of the branch 
of the hyperbola, which does not pass through it, is less than the 
radius of the circle. The polar coordinates (r 3 , 93) of this nearest 
point of the second branch of the hyperbola are given by the 
equations 

tan 93= y/ tan 9 2 , 

r 3 = r 2 cos 92 sec 3 93 . 
34 



— 266 — 

496. When the body is originally placed at one of the points 
of maximum or minimum velocity, without any initial velocity 
along the circle, it remains stationary upon the curve ; but its 
position upon the curve is one of stable equilibrium, when it is 
placed at a point of maximum velocity, and a position of unstable 
equilibrium, when it is placed at a point of minimum velocity. 
When the body is originally placed upon the curve, without any 
initial velocity along the line, at a point different from these points 
of maximum or minimum velocity, it oscillates about that point 
of greatest velocity from which it is not separated by a point of 
least velocity ; its oscillations embrace both the points of great- 
est velocity, when the velocity is sufficient to carry it through 
either of the points of least velocity, that is, when the velocity, 
which corresponds to the initial point in the general equation 
(265 7 ), is less than that which corresponds to one of the points 
of least velocity. When the initial velocity of the body is greater 
than the excess, which is given by equation (265 7 ) of the velocity 
at the initial poitft above the least of the minimum velocities, the 
body constantly moves, in the same direction, through the entire 
circumference. 

497. The case in which the initial velocity of the body is 
just equal to the excess, which is given by equation (265 7 ) of the 
velocity at the initial point above either of the minimum veloc- 
ities, admits of integration. In this case, it is easy to express the 
equation (265 7 ) in the form 

R (D t <pf = 2a 2 k (sin cp — sin Cjp x ) — a 2 R sin 2 ?? (cos 2 (p — cos 2 g^) 
-\- 2 (</ — a 2 pcos p z ) sin r z (cos <p — cos^), 

which by means of (265 20 ) assumes the form 

R (D t cp) 2 = a 2 sm 2p z [2r 2 cos((p -f-<p 2 ) — 2r 2 cos(<jp 1 -]-(p2) 
— R cos 2 (p -\- R cos 2 cp{] . 



— 267 — 

The condition for the determination of the point of min- 
imum velocity gives also the equation 

2r 2 sin (^ -j- y 2 ) = i? sin 2 <p 1} 

which substituted in the previous equation with the notation 

<P=.i\(p — (p 1 ) 

jg- == sin(qp 1 — qp 2 ) 
sin (cjPi + fjPa) 

gives 

{D t <Pf = i a 2 sin 2 \ sin 2 <P [cos 2 ( <£ -f ^ — II] . 

If, therefore, i? is negative and absolutely greater than unity, 
that is, if (p 1 is not in the same quadrant with (p 2 , the value of 
<P is unlimited ; but if II is less than unity, the limits of <£» are 
given by the equation 

cos 2 (<P-\-(p 1 )= II. 
The integral of the equation (267 n ) is 

a t sin p z y/ ( \ cos 2 c/) x — £ iT) 

_ j sin (<£>+?!) y/(cos 2 yi — //) — sin ?1 y/[cos 2 (<p-f yQ — #] 
° cos(0)-|-qr 1 )v / (cos2 9 1 — H) -\- cos ^ \J [cos 2 ( 0> + qn) — -#] " 

498. J7' ^<? rotating line is a parabola, of which the transverse 
axis is vertical, let 

q be the distance from the vertex to the focus of the parabola, 

and let the origin of coordinates be assumed to be upon a level 
with the vertex, and let 

k denote the distance of the vertex from the foot of the perpendicu- 
lar (p) let fall from the origin upon the plane of the parabola. 



— 268 — 

If the axis of x y is the horizontal line, which is drawn in the 
plane of the parabola through its vertex, and if the vertex is 
the origin of x x , the values of z and u are given by the equations 

iqzz=x\, 

u* =^+(* -\-x x f; 

and the equation (262 2 ) is reduced to 

(D. sf = «'/ + « 2 (* + "if + |f + " 



= (*+i)W*f. 



The integral of this equation, in its general form, can be 
obtained by elliptic functions. The point of least velocity along 
the curve is determined by the equation 

2tf(* + »0+!*i = 0j 

but there is no such point, when 

q = — l-. 

* 2a? 

When this latter condition is satisfied, and also 

the velocity of the body along the curve is constant. 
When k vanishes and 

a = a 2 (4 f — p 2 ) -j- 2y^, 
the equation (268 10 ) becomes 

D.1 = ±*S\I (*+$. 



— 269 — 

so that, in this case, the horizontal velocity of the body upon the 
plane of the parabola is constant. 

499. In the especial case, in which the initial velocity is that 
which corresponds to the vanishing of the minimum velocity, let 

x 2 be the value of x x for this point of minimum velocity, 

and the integral of the equation of motion is 

2^v/(« 2 -r-^) = V / (^ 2 i + 4 ? 2 ) + ^ 2 lo g [^-}-v/(^+4 ? 2 )] 

X± — — x% 

500. When the axis (h) of rotation is not vertical, the equation 
of motion is still reduced to the form (261 24 ), and when the rotation 
is uniform, it becomes 

D 2 t s = a 2 u cos" -\-g cos * = I a 2 D s u* -\-g cos*;. 

501. Wlien the rotating line about the inclined axis is straight, if 
the point of the axis of rotation which is nearest to the rotating 
line is assumed as the origin, let 

p be the perpendicular upon the line from the origin, 

let s be counted from the foot of the perpendicular (p), and the 
time from the instant, when the plane of the directions of the axis 
and the rotating line is vertical. The values of u and cos \ are 
given by the equations 

a 2 =p i -j-s 2 sin", 
cos^ = cos * cos * -J- sin h , sin h z cos (« t) ; 



— 270 — 

which reduce the equation (269 18 ) for this case to 

D 2 s = a 2 s sin 2 * -\-g cos * cos * -j- g sin ) sin * cos (a t) . 

The integral of this equation is 

. a t sin * , j-. — a t sin * q cos * cos * q sin * sin * , , . 

s = Ac S +Bc s — y 2 .' 2A /,. ,' . ' cos(al), 

1 or sin , a 2 (1 -j- sin%') v '' 

in which A and Z? are arbitrary constants. 

502. If in the general case of the rotation of a plane curve about the 
inclined axis the time is computed from the instant, when the plane 
of the curve is vertical the expression of (*) is given by the 
formula 

cos * == cos* cos * -\- sin * sin * cos at. 



MOTION OF A BODY IJPOJT A LINE IN OPPOSITION TO FRICTION, OR THROUGH A 

RESISTING MEDIUM. 



503. The forces of nature, which resist the motions of bodies, 
are of various kinds and subject to different laws. While their 
philosophical discussion must be reserved to its appropriate place, 
it is sufficient for the present purpose, to recognize them as forces, 
which are opposed to the motion of bodies, and which depend in 
general upon the relative motions of the body and of the origin 
of the resistance, whether this origin be solid or fluid. 

504. If either of the resisting forces is denoted by JS 1} and 
if (* ) denotes the angle which the direction of its action makes 
with the path of the body, the resistance to the motion of the 
body in its path will be expressed by /Si cos J , which may be 
immediately introduced into the equation of motion. 



— 271 — 

505. If the bod// moves upon a fixed line, the equation of 
motion (243 19 ) becomes 

A/ = Ai2+^ 1 (5 r 1 co B y. 

If there is, likeivise, no motion in the resisting medium, all the 
forces of resistance can be combined in one, which is directly 
opposed to the motion of the body, and the preceding equation 
assumes the form 

D t s' = D s £2 — S. 

506. If there is no external force, these equations become 

D t s' = — S. 

507. The integral of the latter of these equations is 

t= —Js's: 

Let JS have the form 

S=a-\-bs'-\-es' ? , 
in which a and e are positive, in the case of nature, and 

b + \]{4:ae)>0, 

because 8 is always positive when / is positive. The correspond- 
ing integral of (271 17 ) is 

— A 4- 1 W^+A+V ( ^-4ae) 

A 1 ^J^ — iae) °2es'+b — \/(l> 2 —4ae) 

in which A is an arbitrary constant, and the former integral cor- 



— 272 — 

responds to the case of b 2 <^4ae, while the latter corresponds to 
£ 2 >4«e. The velocity vanishes after the time t given by the 
equation 

t = A— ... 2 gv tan 1 - 13 



y/^ae — P) ^(iae — b 2 ) 

^~T v /(46 2 — 4ae) °^b — \/ (b*— 4ae)' 

These values are infinite in form, when 

b 2 = 4:ae; 
but, in this case, the integral is 



ft (6s' + 2 a) I 2 es'-f-S' 

so that the velocity vanishes, when 

t = A + l = A + 



Sj(ae) 



These values become infinite in form when both b and e 
vanish, but, in this case, which includes that of friction upon a 
straight path, the integral is 



t = A — -= s ^^-: 

a a 



and the instant, at which the velocity vanishes is determined by 
the equation 



t 



*0 

° — a' 



When a vanishes, the value of t is actually infinite, so that 
the velocity of the body can never be wholly destroyed by any 
such form of resistance. It would seem, from the preceding equa- 



— 273 — 

tions, that the direction of motion would be reversed after the 
time (7 ). But this conclusion, which is absurd, because it would 
give a resistance the power of creating motion, arises from the 
defective forms of notation which do not express the solution of 
continuity corresponding to the abrupt ceasing of the friction at 
the instant of the suspension of motion. 

508. When the resistance is simply that of friction arising from 
the pressure of the moving body upon the line, to which its motion is 
restricted, let 

p denote the direction of the perpendicular to the fixed line, 
which is drawn in the common plane of the direction of 
the external force and of that of the line, 

dv the elementary angle made by two successive radii of cur- 
vature to the fixed line, and 

a the coefficient of friction, 

and the equation of motion becomes by (245 18 ) 

D t s = DJ2 — aD p £l — ^f= D S S2 — aD p £l — as V . 

509. When there is no external force, this equation becomes 

D t s' = — as'v f ; 
the integral of which is 

log/ = ^4 — av , 

in which A is an arbitrary constant. Another integration gives 
P av — A r •/ ' av — A\ C( av—A\ 

t= l° =JA D » se )=Xy> c > 

in which c is the Naperian base, and ^> the radius of curvature of 
the fixed line. 

35 



— 274 — 

510. If the fixed line is the involute of the circle, and if its 
equation is 

q = Rv, 

the equation (273 28 ) becomes 

t=X(av-l)c— A + B, 

in which B is an arbitrary constant. 

511. If the fixed line is the logarithmic spiral, and if its equa- 
tion is 

Q = Rc , 

the equation (273 28 ) becomes 

a-\-o ' 

in which B is an arbitrary constant. 

512. If the fixed line is the cycloid, and if its equation is 

q = 4 R sin v , 
the equation (273 28 ) becomes 

, 4JR / . \ av — A , „ 

? = j. iasmv — cosj>)c ~r-" 

in which B is an arbitrary constant. 

513. When the resistance of the line is constant, and the resisting 
medium is moving ivith an uniform velocity in an invariable direction, and 
the resistance arising from it is proportional to the velocity in the medium, let 

a be the constant resistance of the line, 

h the resistance of the medium for the unit of velocity, and 

b the velocity of the medium, 



— 275 — 

and if the direction of the motion of the medium is assumed for 
that of the axis of x, the equation of motion becomes 

D,s' = D S S2 — a — h s[ cos * 

= D s £1 — a -f k (b cos ; — /) , 

in which it is carefully to be observed that the sign of a must be 
reversed simultaneously with the direction of motion. 

514. When the fixed line is straight and there is no external force 
the integral of the equation (275 6 ) becomes 

log (s — b cos * -f- j) = A — h t 

in which A is an arbitrary constant. When 

a<^b h cos * , 

the velocity of the body will never be destroyed, but will constantly 
approximate to 

I s a 

But when 

a^>bh cos * , 

the velocity will vanish after the time t , determined by the equation 

log (J — beosU = A — ht . 

If the initial velocity of the body had been negative, the 
equation of motion would have assumed the form 

log(— /if icwi + j) = — 4-fA<; 

so that the velocity would have vanished after the time t , deter- 



— 276 — 

mined by the equation 

log (b cos ' x -f- jj = — A -f- h t . 

The body would then have remained at rest unless the con- 
dition (275 14 ) had been satisfied, in which case its subsequent motion 
would be defined by the equation (275 n ). 

515. When a heavy body moves upon a fixed straight line, and the 
resistances consist of a constant resistance, arising from the friction along 
the line, and also of a resistance arising from a resisting medium, ivhich 
has a uniform motion in the direction of the fixed line ; and when the re- 
sistance of the medium is 'proportional to the square of the velocity of the 
body in the medium, let 

a be the constant of friction, 

b the velocity of the medium, and 

h the resistance of the medium for the unit of velocity. 

The line may be assumed to be vertical without diminishing 
the generality of the investigation and the equation of motion 
will be 

D t s' =zg — a — h(s —by, 

in which the signs of a and h must be reversed simultaneously with 
those of/ and (/ — b) respectively. The equation of motion has 
precisely the same form with that of § 507, so that the forms of 
the integral are the same which are there given, but the constants 
are not subject to the restrictions of that section. 

If, then, the initial velocity is upward and exceeds that of 
the medium, when the medium is also moving upwards, the ascend- 
ing velocity decreases by the law expressed in the equation 

. s>-b = s /^tan[(i-r)^h(g + a))], 



— 277 — 

in which x is an arbitrary constant. This law of ascent continues 
until the body is brought to rest when the medium is not moving 
upwards. But when the medium is moving upwards, it continues 
until the instant (t), when the velocity of the body is the same 
with that of the medium. After this instant, the velocity de- 
creases by the law 

/ -^ = V / ^ Tan[( ^~' r)v/(M ^ + a))]; 

which continues forever if 

g + a<hb* 

and the velocity constantly approximates to that, which is deter- 
mined by the equation 

y-f a = h(s' — bf. 

But when 

g + a>hb 2 , 

the body is brought to a state of rest, in which it continues per- 
manently if 

g — a<ihb 2 . 

But if the motion of the medium is upward, and 
g — a>hb 2 , 

the body moves from the state of rest with an increasing descending 
velocity of which the law is expressed by the equation 

s r -b = yJ^T*n[(i-~* 1 )^(g-a))-], 

in which % x must be determined so that the instant of rest coincides 



— 278 — 

with that given by the equation (277 8 ). The increasing velocity 
continually approximates to that which is determined by the 
equation 

g — a=h(s— bf. 

The state of rest to which the body is brought, when the 
medium is not moving upwards, is permanent if 

a — </>hb 2 . 

But if, on the contrary, 

a — (/<ihb 2 

the body moves from the state of rest with an increasing descending 
velocity, of which the law is expressed by the equation 

when 

in which r 1 must be determined so that the instant of rest coincides 
with that given by the equation (276 3y ). This law of motion con- 
tinues until the instant t 1} when the downward velocity of the 
body becomes the same with that of the medium ; and after this 
instant, the law of increasing velocity of descent is expressed by 
the equation (277 29 ) ; so that the velocity continually approximates 
to that which is determined by the equation (278 4 ). 

But when the body begins to descend from the state of rest, 
and 

9<a, 



— 279 — 
the law of descent is expressed by the equation 

^-h = sJ a -^Goti{r l -t)sJ{h{a-g))-], 

so that the increasing velocity constantly approximates to that 
which is determined by the equation 

a—g = h{s' — bf. 

If the initial velocity is downward, and exceeds that deter- 
mined by the equation (278 4 ), the decreasing velocity when 

g>a 
is expressed by the equation 

in which r is an arbitrary constant. If, therefore, the motion of 
the medium is downward, or if it is upward and the condition 
(277 2 4) is satisfied, the decreasing velocity continually approximates 
to that which is determined by the equation (277 29 ). But if the 
motion of the medium is upward and the condition (277 2i ) is 
satisfied, the body is brought to a state of rest which is permanent 
if the condition (277 n ) is also satisfied. If, however, the condition 
(277 n ) is satisfied by the upward motion of the medium, the body 
leaves the state of rest and ascends with an increasing velocity, 
which is defined by the equation 

/_ r j ==v /.ttf!Cot[0-r 1 )^(AO + a))], 

in which % x must be determined so that the instant of rest coin- 
cides with that which is given by the equation (279 15 ). The 



— 280 — 

ascending velocity continually approximates to that which is 
determined by the equation (277 15 ). 

If the initial velocity is downward, and exceeds that of the 
medium, when the medium is also moving downwards, the de- 
scending velocity, when 

decreases by the law, expressed in the equation 

'-*=v^W(*-ov(*(«-j))]> 

in which % is an arbitrary constant. This law of descent continues 
until the body is brought to rest, when the medium is not moving 
downwards; but when the medium is moving downwards, the 
law continues until the instant t, when the velocity of the body 
is the same with that of the medium. After this instant, the law 
of decreasing velocity becomes 

which continues until the body is brought to rest, when the condi- 
tion (278 9 ) is satisfied. But when, on the contrary, the condition 
(278 12 ) is satisfied, the body continues to move forever with the 
law of decreasing velocity expressed in (280 19 ), and the velocity 
continually approximates to that, which is determined by the 
equation (279 7 ). When the body has been brought to the state 
of rest, the condition and laws of leaving it are the same with 
those defined in (279 23 _ 3J ), when 

g>a. 



281 — 



THE SIMPLE PENDULUM IN A RESISTING MEDIUM. 

516. When the curve is the circumference of a vertical circle, the 
problem is that of the simple pendulum in a resisting medium. If the arc 
of vibration is supposed to be so small that its powers, which are higher 
than the square may be neglected, and if the resistance arising from the 
medium is supposed to be proportional to the velocity, and to be combined 
ivith a constant friction, let 

a be the friction, and 

h the resistance of the medium for the unit of velocity, 

and the equation of motion becomes, by adopting the notation 

of § 487, 

B1 9 = — ^ 9 + a — h D t 9 , 

in which the sign which precedes a, must be the reverse of that of 
D t 9 . The integral of this equation is 



, Ra , qpo — \ht . 7 , 
<P = ±— +jc smkt, 



in which 



It = i/ ^ cos a , 

hh = J ^ sin « , 

and the arbitrary constants have been determined so that the initial 
angular velocity (9^) shall be the maximum velocity, and, therefore, 
the initial value of 9 is 

I Ha 

— 9 

36 



— 282 — 

517. The equation (281 2 i) only applies to the first vibration 
and for the (m-\- l) st vibration, the correct equation is 

in which t m is the instant of the maximum angular velocity {y' m ) of 
that vibration and the doubtful sign is alternately positive and 
negative for the successive oscillations, so that the position of 
maximum velocity is always upon the descending portion of the 
oscillation. 

518. The angular velocity of vibration is expressed by the 
equation 

«/ — «/ „ — h h (* — *m) COS [k (t — T,„) -f ft] 

t — y™ 6 cos« ' 

and it vanishes for the instants 

, — n a 

t — X m + 2k~lc 



which correspond to the beginning and end of the oscillation. The 
whole time of oscillation is, therefore, 

Tn I R 

— T = 7i:\/-seca, 
k V g 

which is invariable, although it exceeds the time of vibration in a vacuum, 
in consequence of the factor, sec a . 

519. The angular deviations of the pendulum from the verti- 
cal at the beginning and end of the oscillation are given by the 
equation 

— R a — , I R (« -J- i n ) tan « 



— 283 — 

whence the whole arc of the (m -\- l) st vibration is 

<P m = 2 (p' m \J — c a a Cos ( I n tan a ) . 

520. The angular deviations of the pendulum from the ver- 
tical at the end of one vibration and the beginning of the next are 
identical, but the deviation from the point of maximum velocity is, 
on account of the change in the position of this point, diminished 
by the quantity 

2Ra 
9 

The successive values of the maximum velocity are therefore 
connected by the equation 

f / (« — i«)tan« o "A 8 —- m > («+**) tana 

fm c \ ~g — T '« + 1 C > 

or 

/ _ / — ft tan a „ IR ■. — (a -\- \ n) tan a 



, , — ft tan a n IR ; 

9« + i = 9 ) mC ~ \ ~g c 



The general expression for the maximum velocity is then 
found to be 

, r — mntancc n IR — (« + hn) tan a (e— m 7r tan a — 1\ 

which, on account of the smallness of a and a, may be reduced to 

/ r — m n tan a ^ I R 

<pm = <foC — 2 mad-. 

The corresponding value of the arc of vibration is 

^=^ og - w ^ aag -i^ g -^ tang Cos(^tan ft ) c " m i tanC ~ 1 1 - 
or 

_ -m n Un a ±maR 
" m "o C ~ • 



— 284 — 

The laiv of the diminution of the arc of vibration and of the maxi- 
mum of velocity is, therefore, such that either of these quantities consists 
of hvo terms, one of which is dependent upon the portion of the resistance, 
which is proportional to the velocity, and decreases in geometrical ratio, 
ivhile the other is principally dependent upon the constant friction and de- 
creases, sensibly, in arithmetical ratio. The vibration ceases when the second 
term of either of these quantities surpasses the first. 

521. If the resistance is proportional to the square of the velocity, 
and if h is its value for the unit of velocity, the equation of the 
motion of the pendulum is 

D 2 (p = — £ s in <p — h (D, <p f . 

If one of the first integrals of this equation is supposed to be 
(254 26 ), in which, however, H is not constant but variable, the 
differential of (254 26 ) gives, by means of this equation and (254 26 ), 

DJI=R 2 D t <pD 2 y-\-gRsmyD t y=z — hR' i {D t yy 

= — 2hD t tp (gRcoscp -j- H), 
D <p II= — 2ghRcos(p — 2hff; 

and the integral of this last equation if 

tan ju, = 2 h, 

is 

II=zAe * l * — g R sin fi sin ((p -j- fi), 

in which A is an arbitrary constant. The equation (254 26 ) is then 
reduced to 

R* {D t 9 ) 2 = 2 A c~ 9 tan ** + 2 g R cos fi cos (9 -f p) ; 

of which the integral is 

, r R 

t T= I 

J<t> 



y/ [2 A c — <p tan 1" -j- 2 g R cos fi cos (9 -J- p) ] ' 



— 285 — 

The signs which precede the quantities h and fi must be re- 
versed in the alternate oscillations. 

522. The angle of greatest deviation from the vertical for 
the (m -j- l)st oscillation is determined by the equation 



g R cos fi 



— w m tan u ( , 

C T COS(y w — fl) 

= c <Pm+1 &nfl cos(cp m+1 -\-p), 



If// is adopted as the sign of finite differences, this equation 
gives, when fi is so small that its square may be neglected, 

J [cos cp m — (sin cp m — (p m cos (p m ) fi]=2 (sin cp m — (p m cos <p m ) fi . 

When the oscillations of the pendulum are so small that the 
fourth power of cp m may be neglected, and also the product of \i 
by (fi n J (p m , this equation is reduced to 

4<p m = — $p<pi; 

of which the approximate integral is 

523. The substitution of (285 5 ) reduces (284 27 ) to the form 



R 



(I) t( pf = cos( ( p-\-fi)-c- ((f + (fm)tanix coa (<p m — fi), 



2 g cos n 

which, when fi is so small that its square may be neglected, becomes 

— (D t (pf = cos (9 + fi) — cos (q> m — fi) + cos <p m (y -f- (p m ) u 

= cos(p — cos (p m — fi [sin (p -\- sin (p m — (9 + cp m ) cos y w ] . 

When the oscillations are very small, this equation may be 



— 286 — 
still further reduced to 

which gives 

The integral of this equation is 

The time of the descending semioscillation, deduced from this 
equation, is 

4-0=^(1+^). 

The time of the preceding semioscillation is obtained by re- 
versing the sign of fji, which gives 

and the time of the ivhole oscillation is, therefore, the same as if the pen- 
dulum vibrated in a vacuum. The preceding formulae and conclusions 
coincide, substantially, with those which are given by Poisson. 

524. If the law of the resistance to the motion of the pendulum 
may be expressed as a function of the time, let 

2T denote the resistance, 

and the motion of the pendulum in a small arc is expressed by 
the formulae (260 9 ) and (261 7 ). If 3" is a periodic function, which 



— 287 — 

has the same period with that of the vibration of the pendulum, 
it may be expressed in the form 

gr=^ +2l ! [^cos( ? 7 v /| + ^)]; 

and, if the variable portions of A sin b and A cos b are denoted by 
d, these equations give 

ff&(Asmb)=k (l-co S (l^))— V^/Jainfc— ^cos^y/l+ft) 

+ ^ cos ft + 1,{^ cos ((z- 1) ^+ ft) 
-^cos^+lj^S+Z^-^cosft], 

ffd(Acosb)= — A sin(rfy/|)— Vy/|cosft— ^sin(2^y/| + ft) 

+ ^ 1 sinft-l I .[^sin(( ? -l)y|4-ft) 
+ ^T-n((/ + l)^ V /| + ft)-^ 1 sinft] ; 

which vanish with zf. 

525. if the vibrations of the pendulum cause the medium to oscil- 
late, the period of the oscillations of the medium is probably the same with 
that of the pendulum, but the successive phases of the motion of the medium 
are likely to lag someivhat behind those of the pendulum. Hence the 
relative velocity of the pendulum to the medium may be ex- 
pressed by the equation 

V=vAco S (t^ + b + (l), 

in which A and b may be regarded as constant for a single vibra- 
tion. 



— 288 — 

If, then, the resistance of the medium is proportional to the 
relative velocity, the value of 9° assumes the form 

er=2AAeofl(yf+a-fp) ; 

and the equations (2878_i 8 ) give 

] ? 2 d(Asmb)=-t s Jj iS m(b + p) 

£ 2 d(Acoab) = — t^Z i co8(b + (i) 

_ i an (2*y/£ + J +/*) + * sin (* + /*), 
whence 

plogil = -.y|cosjJ-i8in(2/ v /|4-2i + /j)+* S in(2J + /?), 

.£« = — y|sin/5— Jcos(2*y/|+2J + |*) + Jcos(2J + /J). 

If T is the time of vibration of the pendulum, the changes 
of A and b in a single vibration are given by the formula 

J\ogA = — ^ry/|cos/9 = — rc-cos/J, 
//j = — *Fi/4sin/J = — 71- sin/3. 

i/^ ^Ae resistance is proportional to the square of the velocity, the 
value of ST assumes the form 

®=2kA 2 + 2kA 2 cos (2^t/|+2^ + 2/5), 

in which the sign of # must be reversed, when the direction of the 



— 289 — 

relative motion of the body to the medium is reversed. This 
value of 9° gives 



k£ 



J_ 
k£ 



^(4sin$) = 2 — 2cos(y|) + cos(y|4-2*4-2/?) 

— h cos (otJ^ + 2^ + 2(i) — I cos (2*4-2/5), 

^(^cos3) = -2sin(y|)- S in( ! 5 v /| + 2J4-2^) 

-ism(3ify/| + 2J + 2^) + | S in(2J + 2/5); 
whence 

- <^ ^1 = 2 sin £ — 2 sin(^t/| -f- i) + sin(^ y/| -f J -f 2 /?) 

— isin(3^i/|4-35-f 2/5) + isin(3J + 2/5)— sin(i-|-2/i), 

^•=2cos5 — 2cos(^y/|4-j)4-cos(^»/|4-i4-2/5) 

— icos(3^y/|4-354-2/3)4-icos(3J4-2/5)— cos(J-f2J). 

The changes of ^4 and # in a vibration are found, by having 
regard to the reversal of the sign of k which corresponds to that 
of V, to be 

(/JA = — -V-£-4 2 cos/5, 
g J b = — -L 6 - It A sin (i . 

i/' - ^e law of the resistance is similar to that of friction so as to 
be constant if the medium is at rest, it must, when the medium is in 
motion, be proportional to the quotient of the relative motion of 
the body through the medium divided by the velocity of the 

37 



hA 



— 290 — 
body. The form of 2T is, then, 

^ acos(*y/§ + ft + /?) 
oob(^| + *) 

in which the sign of a must be reversed, when the direction of the 
relative motion of the body to the medium is reversed. This value 
of 2T gives 

?d(Asmb)= — cos^i/l + ft) — sm (i cos Hog tan(i ^ + ^" 2 +ft ), 

|d (4 cos 0) = — sin (rf y/| + /?) -J- sin 8 sin Hog tan (± re -j- *-^|±_ 5 ) ; 

whence 

*4==-5«ii(f v /} + *.+ /») J 

<J 5 = - ^ oos (^1 + b + /?) - ± sin fi log tan (j * + l M±l) . 

The changes of A and J in a vibration are 
J A = , 

9 
J b = -j- sin fi log tan \ 8 . 

The combination of these values give 

9 9 6 9 

J i = ll sin 8 log tan i8—n- sin 8 — - 1 / * A sin (i . 
^ ° 9 9 

The change of b is exhibited in the motion of the pendulum 



— 291 — 

by a change in the time of vibration, which differs from that which 
it would be in a vacuum. The difference is 

JT= — JbJ- = — -Jb. 
V 9 t 

526. The vibration of the pendulum may be regarded as 
affected by the medium not only in consequence of its direct action 
as resistance, but also indirectly, because a portion of the medium 
may be regarded as composing a part of the moving body, and its 
motion is sustained by the action of gravitation upon the body. 
If, then, 

q denotes the ratio of the mass of that portion of the medium 
which moves with the body to the mass of the body, 

the motion of q may be assumed to have a period identical with 
that of the body, and an amplitude of excursion proportional to 
that of the body, so that its velocity may be of the form 

The resistance, then, arising from the preservation of the 
motion of q may be expressed in 2T by the form 

sr=,z> 1 r=-i^2--n(< v /j+*_/i'). 

The similarity of this form to that of (288 4 ) shows that the 
corresponding influence upon A and b may be expressed by the 
equations 



— 292 — 

The importance of this form of resistance was first noticed 
by Dubuat and has been investigated experimentally by Dubuat, 
Bessel, and Baily. The formulEe (290 27 ) and (291 29 ) may be adopted 
as a guide in the conduct of these and similar investigations. 

527. In the application of the preceding formulae to the re- 
duction of experiments, the quantities a, h, k, and q are inversely 
proportional to the density of the body, and directly proportional 
to the density of the medium, and for bodies of similar forms they 
are nearly in an inverse ratio to their linear dimensions. For 
pendulums of different lengths, h is proportional to the length of 
the pendulum, and h to the time of vibration. If iZj denotes the 
resistance of the medium which is proportional to the velocity for 
the unit of weight and the unit of surface, and if ff 2 denotes the 
resistance which is proportional to the square of the velocity for 
the same unit of weight and surface, the values of h and k, for the 
units of weight and surface, are 

528. The best experiments which have been made with the 
pendulum are almost wholly free from any constant term of resist- 
ance, so that, in their discussion, this term may be neglected which 
reduces the formula (290 26 ) to the form 

JA-= — \ TH X A cos §—%R ff 2 A 2 cos §, 

of which the approximate integral is 

529. In order to illustrate these formulae, they may be ap- 
plied to some of the experiments which have been actually made, 



— 293 — 

and in which the diminution of the arc of vibration has been ob- 
served. For this purpose the observations of Newton, Dubuat, 
Borda, Bessel, and Baily are selected, and the formula (292 28 ) is 
found to be applicable to all these experiments, although the values 
of Hi and H 2 are different for the different experiments. The unit 
of length which is here adopted is the meter, the unit of weight 
is the chiliogramme, and that of time is the mean solar second. 
The measures and weights are, however, given in the form in 
which they were actually observed. 

530. In Newton's first series of experiments upon the dimi- 
nution of the oscillations of a pendulum, a wooden sphere of 
6| English inches in diameter, weighing 57^ ounces, of about 
0.56 specific gravity, and suspended by a fine wire so as to give 
10 2 feet for the length of the pendulum, was vibrated until the arc 
of descent was diminished one fourth or one eighth of its initial 
extent, and the number of vibrations was recorded. From the re- 
duction of these observations, I have obtained for the values of 

#[ = 0.0223 sec/?, 

J7 2 = 0.4473 sec ^. 

In Newton's second series of experiments, a leaden sphere of 
2 inches in diameter, weighing 261 pounds, and suspended so as to 
give 10 £ feet for the length of the pendulum, was vibrated in the 
same way as in the former series. From the reduction of these 
observations, I have obtained 

H t = 0.2044 sec p, 
ff 2 = 0.701 sec /J. 

To test the accuracy of these reductions, and their conformity 
with the given observations, I have computed the lengths of the 



— 294 — 

observed arcs of vibration, and have placed them in the following 
table for comparison. 



COMPARISON OF NEWTON S EXPERIMENTS UPON VIBRATIONS OF THE PENDULUM 

WITH COMPUTATION. 





WOODEN 


SPHERE. 






LEADEN 


SPHERE. 




in 


Computed 


Observed 
-Am 


C—O 


m 


Computed 
■Am 


Observed 
■Ah 


C—O 




in. 


in. 


in. 




in. 


in. 


in. 





64.08 


64 


.08 





64.03 


64 


.03 


n 


56.02 


56 


.02 


30 


56.04 


56 


.04 


22§ 


47.91 


48 


—.09 


70 


47.92 


48 


—.08 





31.86 


32 


—.14 





31.92 


32 


—.08 


18i 


27.92 


28 


—.08 


53 


28.00 


28 


0. 


41§ 


24.19 


24 


.19 


121 


24.07 


24 


.07 





15.99 


16 


—.01 





16.01 


16 


.01 


35i 


14.01 


14 


.01 


901 


13.99 


14 


—.01 


83i 


11.99 


12 


—.01 


204 


11.99 


12 


—.01 





8.04 


8 


.04 





8.05 


8 


.05 


69 


7.01 


7 


.01 


140 


7.01 


7 


.01 


1621 


5.95 


6 


—.05 


318 


5.95 


6 


—.05 





4.01 


4 


.01 





4.03 


4 


.03 


121 


3.50 


' H 


0. 


193 


3.49 


H 


—.01 


272 


2.99 


3 


—.01 


420 


2.97 


3 


—.03 





1.98 


2 


—.02 





2.04 


2 


.04 


164 


1.74 


If 


—.01 


228 


1.74 


If 


—.01 


374 


1.52 


H 


.02 


518 


1.46 


H 


—.04 













1.00 


l 


0. 










226 


.88 


i 

8 


0. 










510 


.75 


f 


0. 



With these values of H x and ff 2 , a minute arc of vibration 
of the wooden sphere would be reduced one eighth part in 446 
vibrations, and one fourth part in 961 vibrations, and a minute 
arc of vibration of the leaden sphere would be reduced one eighth 
part in 290 vibrations, and one fourth part in 625 vibrations. 

531. Dubuat vibrated in water a sphere of 2.645 French 
inches in diameter, weighing in air 40068 grains, and in water 
36448 grains, and suspended so that the length of the pendulum 



— 295 — 

was 36.714 inches ; he observed the arc of descent at each succes- 
sive oscillation. From these observations, I have obtained a result 
which corresponds with his own in respect to the law of diminution 
of oscillation, and which gives for the values of II y and II 2 in water 

ff 2 = 378.7 sec /?. 

Dubuat also vibrated in air a paper sphere of 4.0416 inches in 
diameter, weighing in air 155 grains, with a density 11.33 times as 
great as that of air, and suspended by a fine thread so that the 
length of the pendulum was 36.714 inches. From these observa- 
tions, I have deduced 

£1 = 0, 

H 2 = 0.37 sec §. 

The following table contains the comparison of Dubuat's experi- 
ments with the computations derived from the values of Hi and H % . 

COMPARISON OF DUBUAT's EXPERIMENTS UPON THE DIMINUTION OF THE ARC OF 
VIBRATION OF A PENDULUM WITH COMPUTATION. 



SPHERE IN WATER. 


SPHERE IN AIR. 


Ill 


Computed 


Observed 


C—O 


m 


Computed 
An 


Observed 


C—O 




in. 


in. 


in. 




in. 


in. 


in. 





12.00 


12.00 


0. 





11.90 


12.00 


—.10 


1 


9.21 


9.25 


—.04 


i 


10.10 


10.00 


.10 


2 


7.47 


7.42 


.05 


2 


8.77 


8.70 


.07 


3 


6.28 


6.25 


.03 


3 


7.75 


7.79 


—.04 


4 


5.42 


5.33 


.09 


4 


6.94 


6.96 


—.02 


5 


4.77 


4./5 


.02 










G 


4.25 


4.25 


0. 










7 


3.84 


3.83 


.01 










8 


3.50 


3.48 


.02 










9 


3.22 


3.23 


—.01 










10 


2.97 


2.98 


—.01 











— 296 — 

532. Borda vibrated a platinum sphere of 16i lines in diameter, 
weighing with the wire and screw 9963 grains, and suspended by a 
wire so that the length of the pendulum was 3.95497 metres. These 
observations give for the values of H x and H 2 in air 

#i = 0.10722 sec /?, 
# 2 =0.6267 sec /?. 

In his observations for determining the length of the seconds 
pendulum, this same pendulum was vibrated by Borda, and the 
lengths of its arcs of vibration were observed. From the mean of 
these observations, I have obtained the values of 1^ and II 2 , 

# x = 0.11214 sec/?, 
II 2 = 0.6564 sec/?. 

Borda vibrated the same sphere with a smaller wire, so that 
the weight was reduced to 9958 grains, and the length increased to 
3.95597 metres. From these observations I have derived 

#i = 0.1134 sec/?, 
# 2 = 0.590 sec/?. 

The comparison of Borda's experiments with the computations 
based upon these values of #i and H 2 is contained in the following 
tables. 



COMPARISON OF BORDA's OBSERVATIONS UPON THE DIMINISHED VIBRATIONS OF 
THE PENDULUM WITH COMPUTATION. 

First Experiment with direct reference to the Diminution of the Arc of Vibration. 



m 


Computed 

An 


Observed 
An 


C—O 


m 


Computed 


Observed 
An 


C—O 





12<)'o 


120^0 


0. 


12600 


4.2 


4.1 


/ 

0.1 


1800 


61.2 


61.2 


0. 


14400 


2.8 


2.7 




3600 


35.6 


35.4 


.2 


16200 


1.9 


1.8 




5400 


22.1 


21.9 


.2 


18000 


1.3 


1.2 




7200 


14.2 


14.1 


.1 


19800 


0.9 


0.8 




9000 


9.4 


9.4 


0. 


21600 


0.6 


0.5 




10800 


6.2 


6.3 


—.1 


36000 


0.002 


Very minute. 





— 297 



Experiments for determining the Length of the Second's Pendulum ivith the Pendulum used in the First 

Experiment. 



m 

Mean Value. 


Computed 
An 


Observed 
A m 


C—O 


Computed 


Observed 
A m 


C—O 


Computed 


Observed 

A, 


C—O 





64' 


&( 


/ 




d 


67 y 


/ 




<oi 


d 





2169 


321 


32 


i 

2 


34 


34 





32 


32 





4338 


18 


19 


—1 


181 


19 


1 

2 


18 


18 





6507 


101 


11 


I 
2 


11 


11 





101 


11 


2 


8676 


6| 


7 


1 
2 


6* 


7 


1 

2 


6 


6 








60 


60 





61 


61 





64J, 


64* 





2169 


31 


31 





311 


311 





32. V 


32* 





4338 


17* 


17 


X 

2 


17* 


18 


X 

2 


18 


17* 


X 

2 


6507 


10 


10 





10 


10 





10* 


10 


1 

2 


8676 








6 


6 





6* 


6 


X 

2 





63 


63 





68 


68 





61 


61 





2169 


32 


32 





34 


341 


1 

2 


31* 


31* 





4338 


18 


18 





19 


191 


1 

2 


17* 


17 


1 

2 


6507 


101 


10 


1 

2 


11 


11* 


1 

2 


10 


10 





8676 


6 


H 


X 

4 


61 


7 


-* 


6 


6 






2169 


591 
301 


591 
301 






571 
30 


571 
30 






62 
31* 


62 
31* 






4338 


17 


17 





17 


17 





18 


17 


1 


6507 


10 


10 





H 


10 


— * 


10* 


10 


X 

2 


8676 








6 


6 





6 


6 








67 


67 





65 


65 





63 


63 





2169 


34 


34 





33 


331 


— i 


32 


32 





4338 


181 


19 


1 

2 


181 


18| 





18 


17* 


X 

2 


6507 


11 


11 





101 


11" 


— 1 


10* 


10 


JL 
2 


8676 


H 


6 


X 

2 


61 


6* 





6 


6 








71 


71 





591 


591 











2169 


35 


341 


X 

2 


31 


31 











4338 


191 


19 


X 

2 


171 


17 


* 








6507 


11 


11 





10 


10 











8676 


7 


7 





6 


6 












Experiments J 


or determining the Length of the Second' 


s Pendulum with the Second Pendulum. 


m 


Comp'd 
An 


Observ'd 

A n 


C-0 


m 


Comp'd 

A m 


Observ'd 
An 


C—O 


m 


Computed 

A m 


Observed 

A m 


C-0 




/ 


/, 


/ 




/ 


J 


/ 


j 


/ 


1 


/ 





551 


551 








79 


79 








111 


110 


1 


1575 


341 


35 


1 

2 


1538 


47 


47 





1445 


641 


641 





3150 


221 


23 


1 

2 


3114 


30 


30 





2970 


401 


40 


1 

2 


4725 


15 


16 


—1 


4690 


191 


20 




4495 


26 


26 





6300 


10 


101 


1 
2 


6266 


13 


14 


—1 


6020 


171 


18 


1 

2 


7875 


7 


71 


1 
2 


7842 


9 


91 


1 

2 


7545 


12 


12 





9450 


5 


5 


9418 


5 


6* 


1 

2 


9070 


8 


8* 


2 



38 



— 298 — 

533. In Bessel's experiments made for the determination of 
the length of the second's pendulum of Konigsberg, a brass sphere 
of 24.164 lines in diameter, weighing 0^.695364 was suspended so 
that the length of the pendulum was 1305.3 lines. From his ob- 
servations with this pendulum, I have found these values of 1^ 
and II 2 . 

H x = 0.05698 sec /?, 

.#2=0.529sec/?. 

The same sphere was also vibrated with a length of pendulum 
of 441.8 lines, from the observations of which I have deduced 

.#!=: 0.0452 sec/?, 
H 2 =0.587 sec /?. 

Bessel also vibrated an ivory sphere, weighing 0M5112, and 
having a diameter of 24.094 lines, with each of the preceding 
lengths of pendulum. From his observations with this sphere and 
the long pendulum, I have obtained 



H 1 = 0.05517 sec/?, 
JI 2 =0.512 sec/2; 



and from his observations with the short pendulum, 

7^=0.0509 sec/?, 
i7 2 =0.282 sec/?. 

In Bessel's experiments for the determination of the length of 
the second's pendulum at Berlin, a hollow cylinder was vibrated, 
of which the diameter of the base was 15.305 lines, and the altitude 



— 299 — 

15.296 lines, weighing, with its appendages, when it was filled with 
lead, A '.67920, and when it was empty, 0\22595. It was suspended 
in two different modes, in one of which the length of the pen- 
dulum was 1304.8 lines, when the cylinder was filled, and 1303.8 
lines, when it was empty ; and, in the other mode of suspension 
the length was 440.9 lines when the cylinder was filled, and 
440.7 lines when it was empty. From his observations with this 
pendulum, I have obtained the following values of 1^ and II 2 . 

When the cylinder was full, and the suspension was long, the 
values were 

#!= 0.08544 sec/?, 

H 2 = 0.733 sec /? ; 

when it was full, and the suspension short, they were 

^ = 0.07026 sec/?, 
II 2 = 0.724 sec/?. 

When the cylinder was empty, and the suspension long, the 
values were 

ff= 0.09578 sec /?, 

11= 0.559 sec /? ; 

when it was empty, and the suspension short, they were 

^ = 0.07003 sec/?, 
# 2 =0.270 sec/?. 

In order to compare the theory of these values with ex- 
periment, all the values of observation have been recomputed, 
and the comparisons are contained in the following tables. 



300 — 



COMPARISON OF BESSEL S OBSERVED ARCS OF VIBRATION OF THE PENDULUM WITH 

THE COMPUTED ARCS. 



1. Experiments witk ike Brass Sphere and Long Suspension. 



m 


Computed 
■A-m 


Observed 


C—O 


Computed 


Observed 
■Am 


C—O 


Computed 

An 


Observed 
-A-m 


C—O 





38.3 


38.3 





39.0 


39.0 





39.5 


39.5 





500 


33.7 


33.8 


—.1 


34.2 


34.2 





34.6 


34.6 





1000 


29.7 


29.8 


—.1 


30.2 


30.1 


.1 


30.5 


30.5 





1500 


26.4 


26.4 





26.8 


26.8 





27.1 


26.8 


.3 


2000 


23.5 


23.6 


—.1 


23.9 


23.8 


.1 


24.1 


23.9 


.2 


2500 


21.0 


20.9 


.1 


21.3 


21.3 





21.6 


21.6 





3000 


18.8 


18.8 





19.1 


19.2 


—.1 


19.3 


19.3 





3500 


16.9 


16.9 





17.2 


17.2 





17.4 


17.3 


—.1 


4000 


15.3 


15.4 


—.1 


15.5 


15.5 





15.7 


15.7 








39.7 


39.9 


—.2 


39.0 


39.3 


—.3 


39.6 


39.7 


—.1 


500 


34.8 


34.6 


.2 


34.2 


34.1 


.1 


34.7 


34.8 


—.1 


1000 


30.7 


30.4 


.3 


30.2 


30.0 


.2 


30.6 


30.5 


.1 


1500 


27.2 


27.1 


.1 


26.8 


26.4 


.4 


27.1 


26.9 


.2 


2000 


24.2 


24.1 


.1 


23.9 


23.5 


.4 


24.2 


24.0 


.2 


2500 


21.6 


2*1.5 


.1 


21.3 


20.9 


.4 


' 21.6 


21.4 


.2 


3000 


19.4 


19.3 


.1 


19.1 


18.5 


.6 


19.4 


19.3 


.1 


3500 


17.4 


17.3 


.1 


17.2 


16.4 


.8 


17.4 


17.3 


.1 


4000 


15.7 


15.5 


.2 


15.5 


14.6 


.9 


15.7 


15.5 


.2 





38.6 


38.6 





40.0 


40.3 


—.3 


40.1 


39.9 


.2 


500 


33.9 


33.9 





35.1 


34.9 


.2 


35.1 


35.2 


1 


1000 


29.9 


29.9 





30.9 


30.8 


.1 


31.0 


31.0 





1500 


26.5 


26.6 


—.1 


27.4 


27.2 


.2 


27.4 


27.5 


1 


2000 


23.7 


23.6 


.1 


24.4 


24.2 


.2 


24.4 


24.4 





2500 


21.1 


21.2 


—.1 


21.8 


21.8 





21.8 


21.9 


— 1 


3000 


19.0 


19.0 





19.5 


19.5 





19.6 


19.6 





3500 


17.1 


17.1 





17.5 


17.4 


.1 


17.6 


17.6 





4000 


15.4 


15.4 





15.8 


15.6 


.2 


15.8 


15.9 


— 1 





39.1 


39.1 





39.3 


39.2 


.1 


38.8 


38.5 


.3 


500 


34.3 


34.3 





34.5 


34.5 





34.0 


34.0 





1000 


30.3 


30.3 





30.4 


30.5 


—.1 


30.1 


30.2 


— 1 


1500 


26.9 


26.9 





27.0 


27.1 


—.1 


26.7 


27.0 


— 3 


2000 


23.9 


23.9 





24.0 


24.2 


—.2 


23.8 


24.0 


— 2 


2500 


21.4 


21.4 





21.5 


21.8 


—.3 


21.3 


21.5 


— 2 


3000 


19.2 


19.3 


—.1 


19.3 


19.4 


—.1 


19.1 


19.3 


—.2 


3500 


17.2 


17.3 


—.1 


17.3 


17.5 


—.2 


17.2 


17.4 


—.1 


4000 


15.5 


15.6 


—1 


15.6 


15.7 


1-1 


15.5 


15.5 






— 301 — 





1. Experiments wi 


th the Brass Sphere 


and Long 


Suspension. — Continued. 




m 


Computed 
A m 


Observed 
-A m 


C—O 


Computed 
■Am 


Observed 

A m 


C—O 


Computed 

A m 


Observed 

-4, 


C—O 





39.1 


39.1 





37.8 


37.7 


.1 


39.6 


39.7 


—.1 


500 


34.3 


34.2 


.1 


33.2 


33.3 


—.1 


34.7 


34.7 





1000 


30.3 


30.2 


.1 


29.4 


29.3 


.1 


30.6 


30.6 





1500 


26.9 


27.0 


—.1 


26.1 


26.2 


—.1 


27.1 


27.1 





2000 


23.9 


24.0 


—.1 


23.2 


23.4 


—.2 


24.2 


24.1 


.1 


2500 


21.4 


21.5 


—.1 


20.8 


20.9 


—.1 


21.6 


21.6 





3000 


19.2 


19.2 





18.7 


18.7 





19.4 


19.4 





3500 


17.2 


17.3 


—.1 


16.8 


16.7 


.1 


17.4 


17.3 


.1 


4000 


15.5 


15.5 





15.1 


15.2 


—.1 


15.6 


15.6 


.1 





39.0 


39.0 





41.7 


41.6 


.1 


39.4 


39.4 





500 


34.2 


34.1 


.i 


36.5 


36.6 


—.1 


34.6 


34.5 


.1 


1000 


30.2 


30.1 


.i 


32.1 


32.3 


—.2 


30.5 


30.4 


.1 


1500 


26.8 


26.5 


.3 


28.4 


28.6 


—.2 


27.0 


27.1 


—.1 


2000 


23.9 


24.0 


—.1 


25.3 


25.5 


—.2 


24.1 


24.1 





2500 


21.3 


21.4 


—.1 


22.5 


22.7 


—.2 


21.5 


21.6 


—.1 


3000 


19.1 


19.2 


—.1 


20.2 


20.3 


—.1 


19.3 


19.4 


—.1 


3500 


17.2 


17.2 





18.1 


18.1 





17.3 


17.3 





4000 


15.5 


15.5 





16.3 


16.3 





15.6 


15.6 








39.2 


39.4 


—.2 


38.6 


38.6 





38.5 


39.3 


—.8 


500 


34.4 


34.4 





33.9 


33.4 


.5 


33.8 


34.2 


—.4 


1000 


30.3 


30.3 





29.9 


29.9 





29.8 


30.0 


—.2 


1500 


26.9 


26.9 





26.5 


26.7 


— 2 


26.5 


26.3 


.2 


2000 


24.0 


24.0 





23.7 


23.6 


.1 


23.6 


23.2 


.4 


2500 


21.4 


21.5 


— 1 


21.1 


21.4 


— 3 


21.1 


20.5 


.6 


3000 


19.2 


19.2 





19.0 


19.2 


— 2 


18.9 


18.3 


.6 


3500 


17.2 


17.1 


.1 


17.1 


17.2 


— 1 


17.0 


16.3 


.7 


4000 


15.6 


15.4 


.2 


15.4 


15.4 





15.3 


14.5 


.8 





40.0 


39.9 


.1 


39.9 


39.6 


.3 


39.3 


39.0 


.3 


500 


35.1 


34.9 


.2 


35.0 


35.0 





34.5 


34.5 





1000 


30.9 


30.9 





30.8 


30.9 


— 1 


30.4 


30.5 


— 1 


1500 


27.4 


27.5 


—.1 


27.3 


27.5 


— 2 


27.0 


27.1 


— 1 


2000 


24.4 


24.4 





24.3 


24.3 





24.0 


24.1 


— 1 


2500 


21.8 


21.9 


—.1 


21.7 


21.7 





21.5 


21.5 





3000 


19.5 


19.7 


— 2 


19.5 


19.5 





19.3 


19.3 





3500 


17.5 


17.6 


—.1 


17.5 


17.3 


.2 


17.3 


17.3 





4000 


15.8 


15.8 





15.8 


15.5 


.3 


15.6 


15.5 


.1 





39.7 


39.8 


—.1 


38.9 


38.8 


.1 


38.7 


38.7 





500 


34.8 


34.8 





34.1 


34.0 


.1 


34.0 


34.3 


—.3 


1000 


30.7 


30.7 





30.1 


30.2 


— 1 


30.0 


29.9 


.1 


1500 


27.2 


27.2 





26.7 


26.9 


— 2 


26.6 


26.4 


.2 


2000 


24.2 


24.2 





23.8 


24.0 


—.2 


23.7 


23.5 


.2 


2500 


21.7 


21.8 


—.1 


21.3 


21.4 


—.1 


21.2 


•21.1 


.1 


3000 


19.4 


19.4 





19.1 


19.2 


—.1 


19.0 


18.9 


.1 


3500 


17.4 


17.4 





17.2 


17.2 





17.1 


16.8 


.3 


4000 


15.7 


15.6 


.1 


15.5 


15.4 


.1 


15.4 


15.2 


.2 



— 302 





1. Experiments with the Brass Sphere 


and Long Suspension. — Continued. 




m 


Computed 
■Am 


Observed 
An 


C—O 


Computed 

A* 


Observed 
An 


C—0 


Computed 


Observed 

Ai 


C—0 





38.7 


38.7 





39.3 


39.3 





39.1 


39.2 


—.1 


500 


34.0 


34.1 


—.1 


34.5 


34.7 


—.2 


34.3 


34.2 


.1 


1000 


30.0 


30.0 





30.4 


30.2 


.2 


30.3 


30.3 





1500 


26.6 


26.6 





27.0 


27.0 





26.9 


27.0 


—.1 


2000 


23.7 


23.6 


.1 


24.0 


24.1 


—.1 


23.9 


23.9 





2500 


21.2 


21.2 





21.5 


21.5 





21.4 


21.4 





3000 


19.0 


19.0 





19.3 


19.3 





19.2 


19.3 


—.1 


3500 


17.1 


16.9 


.2 


17.3 


17.3 





17.2 


17.2 





4000 


15.4 


15.3 


.1 


15.6 


15.5 


.1 


15.5 


15.5 








39.0 


39.0 





39.8 


39.7 


.1 








500 


34.2 


34.1 


.1 


34.9 


34.9 











1000 


30.2 


30.1 


.1 


30.8 


30.8 











1500 


26.8 


26.8 





27.3 


27.2 


.1 








2000 


23.9 


23.7 


.2 


24.3 


24.3 











2500 


21.3 


21.2 





21.7 


21.7 











3000 


19.1 


19.2 


—.1 


19.4 


19.4 











3500 


17.2 


17.2 





17.5 


17.4 


.1 








4000 


15.5 


15.4 


.1 


15.7 


15.6 


.1 











2. 


Experiments with the Brass Sphere and the Short 


Suspension 






m 


Computed 
-A-m 


Observed 


C—O 


Computed 
■A m 


Observed 
An. 


C—O 


Computed 
Am 


Observed 
An 


C—O 





14.4 


14.65 


—.2 


13.2 


13.5 


—.3 


12.4 


12.4 





560 


13.5 


13.7 


—.2 


12.4 


12.7 


—.3 


11.7 


11.6 


.1 


1120 


12.7 


12.8 


—.1 


11.7 


11.9 


—.2 


11.0 


10.9 


.1 


1680 


12.0 


11.9 


.1 


11.0 


11.0 


.0 


10.3 


10.2 


.1 


2240 


11.3 


11.0 


.3 


10.4 


10.3 


.1 


9.7 


9.6 


.1 


2800 


10.6 


10.3 


.3 


9.7 


9.7 





9.2 


9.0 


.2 


3360 


10.0 


9.6 


.4 


9.2 


9.0 


.2 


8.6 


8.5 


.1 


3920 


9.4 


8.9 


.5 


8.6 


8.4 


.2 


8.1 


8.0 


.1 


4480 


8.8 


8.3 


.5 


8.1 


7.9 


.2 


7.6 


7.5 


.1 


5040 


8.3 


7.8 


.5 


7.6 


7.4 


.2 


7.2 


7.1 


.1 


5600 


7.8 


7.3 


.5 


7.2 


7.0 


.2 


6.8 


6.7 


.1 





12.2 


12.3 


—.1 


11.5 


11.6 


—.1 


12.2 


12.2 





560 


11.5 


11.5 





10.9 


10.9 





11.5 


11.5 





1120 


10.8 


10.8 





10.3 


10.3 





10.8 


10.9 


— 1 


1680 


10.2 


10.1 


.1 


9.7 


9.7 





10.2 


10.3 


— 1 


2240 


9.6 


9.5 


.1 


9.1 


9.1 





9.6 


9.7 


—.1 


2800 


9.0 


8.9 


.1 


8.6 


8.6 





9.0 


9.1 


—.1 


3360 


8.5 


8.4 


.1 


8.1 


8.15 


—.1 


8.5 


8.5 





3920 


8.0 


8.0 





7.6 


7.7 


—.1 


8.0 


8.1 


—.1 


4480 


7.5 


7.5 





7.1 


7.3 


—.2 


7.5 


7.7 


—.2 


5040 


7.1 


7.0 


.1 


6.7 


6.9 


—.2 


7.1 


7.2 


—.1 


5600 


6.7 


6.5 


.2 


6.3 


6.4 


—.2 


6.7 


6.8 


—.1 



303 — 





2. Expert 


ments with the Bras 


: Sphere and the Short Suspei 


>sion. — Continued. 




m 


Computed 

A. 


Observed 


C—O 


Computed 


Observed 
4 m 


C—O 


Computed 


Observed 

An 


C—O 





12.5 


12.3 


.2 


12.8 


12.7 


.1 


12.9 


12.8 




560 


11.8 


11.7 


.1 


12.0 


11.95 


.1 


12.1 


12.0 




1120 


11.1 


11.0 


.1 


11.3 


11.3 





11.4 


11.3 




1680 


10.4 


10.4 





10.7 


10.7 





10.8 


10.7 




2240 


9.8 


9.8 





10.0 


10.15 


—.1 


10.1 


10.2 


— .1 


2800 


9.2" 


9.2 





9.5 


9.5 





9.5 


9.7 


—.2 


3369 


8.7 


8.75 


—.1 


8.9 


8.9 





9.0 


9.1 


—.1 


3920 


8.2 


8.3 


—.1 


8.4 


8.4 





8.4 


8.6 


—.2 


4480 


7.7 


7.9 


—.2 


7.9 


8.0 


—.1 


7.9 


8.1 


—.2 


5040 


7.2 


7.45 


—.2 


7.4 


7.6 


—.2 


7.5 


7.7 


—.2 


5600 


6.8 


7.0 


—.2 


7.0 


7.2 


—.2 


7.0 


7.2 


—.2 





13.0 


12.9 


.1 


10.9 


10.9 





13.4 


13.2 


.2 


560 


12.2 


12.1 


.1 


10.4 


10.3 


.1 


12.6 


12.5 


.1 


1120 


11.5 


11.4 


.1 


9.7 


9.7 





11.9 


11.8 


.1 


1680 


10.8 


10.8 





9.2 


9.2 





11.2 


11.2 





2240 


10.2 


10.3 


—.1 


8.6 


8.7 


—.1 


10.5 


10.6 


—.1 


2800 


9.6 


9.8 


—.2 


8.1 


8.2 


—.1 


9.9 


9.9 





3360 


9.0 


9.2 


—.2 


7.6 


7.7 


—.1 


9.3 


9.3 





3920 


8.5 


8.8 


—.3 


7.2 


7.2 





8.8 


8.8 





4480 


8.0 


8.2 


—.2 


6.8 


6.8 





8.3 


8.3 





5040 


7.5 


7.8 


—.3 


6.4 


6.4 





7.8 


7.85 


—.1 


5600 


7.1 


7.4 


—.3 


6.0 


6.0 





7.3 


7.45 


—.1 





13.3 


13.3 





11.1 


11.3 


—.2 


12.4 


12.5 


—.1 


560 


12.5 


12.5 





10.5 


10.5 





11.7 


11.7 





1120 


11.8 


11.8 





9.9 


9.8 


.1 


11.0 


10.9 


.1 


1680 


11.1 


11.1 





9.3 


9.3 





10.3 


10.2 


.1 


2240 


10.4 


10.5 


—.1 


8.8 


8.8 





9.7 


9.6 


.1 


2800 


9.8 


9.8 





8.3 


8.3 





9.2 


9.0 


.2 


3360 


9.2 


9.2 





7.8 


7.8 





8.6 


8.5 


.1 


3920 


8.7 


8.7 





7.3 


7.2 


.1 


8.1 


8.0 


.1 


4480 


8.2 


8.2 





6.9 


6.8 


.1 


7.6 


7.6 





5040 


7.7 


7.7 





6.5 


6.4 


.1 


7.2 


7.2 





5600 


7.3 


7.3 





6.1 


6.1 





6.8 


6.8 








11.7 


11.8 


—.1 














560 


11.1 


11.1 

















1120 


10.4 


10.4 

















1680 


9.8 


9.8 

















2240 


9.3 


9.2 


.1 














2800 


8.7 


8.7 

















3360 


8.2 


8.2 

















3920 


7.7 


7.7 

















4480 


7.3 


7.3 

















5040 


6.8 


6.8 

















5600 


6.4 


6.4 


















— 304 — 







3. Experiments with the Ivory 


Sphere and Long Suspension. 






m 


Computed 
■A-m 


Observed 
Am 


C—O 


Computed 
A m 


Observed 

A m 


C— 


Computed 
A m 


Observed 

A m 


C—O 





36.5 


3C.4 


.1 


38.9 


38.9 





38.7 


38.6 


.1 


500 


21.5 


22.0 


—.5 


22.7 


22.7 





22.6 


22.7 


—.1 


1000 


13.5 


13.2 


.3 


14.2 


14.3 


—.1 


14.1 


14.3 


—.2 





38.9 


38.9 





37.9 


37.8 


.1 


37.9 


37.9 





500 


22.7 


22.9 


—.2 


22.2 


22.6 


—.4 


22.2 


22.4 


—.2 


1000 


14.2 


14.5 


—.3 


13.9 


14.3 


—.4 


13.9 


14.0 


—.1 





39.1 


39.2 


—.1 


37.4 


37.5 


—.1 


38.5 


38.5 





500 


22.7 


22.4 


.3 


21.9 


21.7 


.2 


22.5 


22.3 


.2 


1000 


14.2 


13.7 


.5 


13.7 


12.9 


.8 


14.0 


14.2 


—.2 





38.4 


38.4 





37.0 


37.1 


—.1 


37.3 


37.3 





500 


22.4 


22.0 


.4 


21.7 


21.1 


.6 


21.9 


21.8 


.1 


1000 


14.0 


14.0 





13.6 


13.4 


.2 


13.7 


13.9 


—.2 





37.2 


37.3 


—.1 


36.8 


36.8 





37.1 


36.9 


.2 


500 


21.8 


21.7 


.1 


21.6 


21.7 


—.1 


21.8 


22.1 


—.3 


1000 


13.7 


13.8 


—.1 


13.6 


13.4 


.2 


13.7 


13.9 


—.2 





34.7 


34.7 

















500 


20.5 


20.6 


—.1 














1000 


13.3 


13.0 


.3 



















1. Experiments with the Ivory 


Sphere and Short Suspension. 






m 


Computed 

A m 


Observed 

A- m 


C—O 


Computed 

A m 


Observed 

A m 


C—O 


Computed 

A m 


Observed 

A m 


C—O 





12.3 


12.3 





13.6 


13.6 





13.9 


14.0 


—.1 


650 


9.3 


9.3 





10.1 


10.0 


.1 


10.3 


10.1 


.2 


1300 


7.1 


7.2 


—.1 


7.6 


7.8 


—.2 


7.8 


7.8 





1950 


5.4 


5.7 


—.3 


5.8 


5.9 


—.1 


5.9 


5.8 


.1 


2600 


4.2 


4.3 


—.1 


4.4 


4.3 


.1 


4.5 


4.3 


.2 





13.0 


13.1 


—.1 


14.8 


14.9 


—.1 


14.3 


14.3 





650 


9.9 


9.9 





10.9 


10.9 





10.6 


10.7 


—.1 


1300 


7.5 


7.5 





8.2 


8.0 


.2 


8.0 


8.0 





1950 


5.7 


5.7 





6.2 


6.0 


.2 


6.0 


6.0 





2600 


4.5 


4.5 





4.8 


4.5 


.3 


4.6 


4.6 








12.9 


13.1 


— .2 


14.0 


14.0 





J3.2 


13.0 


.2 


650 


9.6 


9.6 





10.4 


10.4 





19.8 


19.9 


—.1 


1300 


7.2 


7.0 


.2 


7.8 


8.0 


—.2 


7.3 


7.4 


—.1 


1950 


5.5 


5.4 


.1 


5.9 


6.1 


—.2 


5.6 


5.9 


—.3 


2600 


4.2 


4.1 


.1 


4.5 


4.5 





4.3 


4.4 


—.1 





13.3 


13.1 


.2 


16.0 


16.0 





16.8 


16.8 





650 


9.8 


10.0 


—.2 


11.8 


11.8 





12.4 


12.5 


—.1 


1300 


7.4 


7.3 


.1 


8.9 


8.8 


.1 


9.3 


9.4 


—.1 


1950 


5.6 


5.8 


—.2 


6.7 


6.8 


—.1 


7.1 


7.1 





2600 


4.3 


4.5 


—.2 


5.2 


5.2 





5.4 


5.5 


—.1 



305 



4. Experiments with the Ivory Sphere and Short Suspension. — Continued. 



m 


Computed 


Obserred 
-4. 


C—O 


Computed 
An 


Observed 
An 


C—O 


Computed 

A m 


Observed 


C—O 





16.6 


16.6 





17.8 


18.0 


—.2 


16.3 


16.3 





650 


12.2 


12.1 


.1 


13.0 


13.0 





12.0 


12.2 


—.2 


1300 


9.2 


9.2 





9.7 


9.5 


.2 


9.0 


9.1 


—.1 


1950 


7.0 


7.0 





7.3 


7.1 


.2 


6.8 


.7.0 


—.2 


2G00 


5.3 


5.5 


—.2 


5.6 


5.5 


.1 


5.2 


5.2 








16.1 


16.0 


.1 














650 


11.8 


12.0 


.2 














1300 


8.8 


9.0 


—.2 














1950 


6.7 


6.8 


—.1 














2600 


5.1 


5.1 


















5. Experiments with the Full Cylinder and Long Suspension. 



m 


Computed 
An 


Observed 


C—O 


Computed 
An 


Observed 

■A m 


C—O 


Computed 
■A- m 


Observed 
A m 


C—O 





39.8 


39.8 





41.5 


41.2 


.3 


38.3 


38.9 


—.6 


500 


35.8 


35.9 


—.1 


37.3 


37.5 





34.5 


34.3 


.2 


1000 


32.4 


32.2 


.2 


33.6 


33.8 


—.2 


31.2 


30.8 


.4 


1500 


29.4 


29.4 





30.5 


30.7 


— .2 


28.4 


27.8 


.6 


2000 


26.7 


26.6 


.1 


27.7 


27.8 


—.1 


25.8 


25.2 


.6 


2500 


24.3 


24.3 





25.2 


25.3 


—.1 


23.6 


22.9 


.7 


3000 


26.3 


22.0 


.3 


23.0 


23.2 


—.2 


21.6 


20.6 


1.0 


3500 


20.4 


20.3 


.1 


21.1 


21.3 


— .2 


19.8 


18.7 


1.1 


4000 


18.7 


19.0 


— 3 


19.3 


19.5 


—.2 


18.2 


17.1 


1.1 





39.4 


39.6 


— 2 


41.0 


41.5 


—.5 


41.7 


41.8 


—.1 


500 


35.5 


35.3 


.2 


36.9 


36.9 





37.5 


37.6 


—.1 


1000 


32.1 


31.9 


.2 


33.3 


32.9 


.4 


33.8 


33.5 


.3 


1500 


29.1 


29.0 


.1 


30.1 


29.9 


.2 


30.6 


30.5 


.1 


2000 


26.5 


26.2 


.3 


27.4 


27.1 


.3 


27.8 


27'.7 


.1 


2500 


24.1 


24.0 


.1 


25.0 


24.6 


.4 


25.3 


25.4 


—.1 


3000 


22.1 


22.0 


.1 


22.8 


22.5 


.3 


23.1 


23.2 


—.1 


3500 


20.2 


20.0 


..2 


20.9 


20.6 


.3 


21.2 


21.2 





4000 


18.6 


18.3 


.3 


19.1 


19.0 


.1 


19.4 


19.3 


.1 





39.5 


39.2 


.3 


40.2 


40.3 


—.1 


42.7 


42.7 





500 


35.6 


35.6 





36.2 


36.1 


.1 


38.3 


38.1 


.2 


1000 


32.1 


32.4 


—.3 


32.7 


32.7 





34.5 


34.6 


—.1 


1500 


29.2 


29.4 


—.2 


29.6 


30.0 


—.4 


31.2 


31.4 


— .2 


2000 


26.5 


26.6 


—.1 


26.9 


27.1 


—.2 


28.4 


28.3 


.1 


2500 


24.2 


24.3 


—.1 


24.5 


24.7 


— .2 


25.8 


25.9 


—.1 


3000 


22.1 


22.3 


.2 


22.4 


22.6 


—.2 


23.6 


23.6 





3500 


20.3 


20.5 


—.2 


20.6 


20.7 


—.1 


21.6 


21.6 





4000 


18.6 


18.7 


—.1 


18.9 


18.9 





19.8 


19.9 


—.1 



39 



306 — 



5. Experiments with the Full Cylinder and Long Suspension. — Continued. 



m 


Computed 


Observed 


C—O 


Computed 


Observed 
An 


C—O 


Computed 
An 


Observed 


C—O 





42.3 


42.5 


—.2 


43.2 


43.1 


.1 


42.0 


41.8 


.2 


500 


38.0 


37.9 


.1 


38.8 


38.8 





37.7 


38.4 


—.7 


1000 


34.2 


34.0 


.2 


34.9 


35.0 


—.1 


34.0 


34.0 





1500 


31.0 


31.1 


—.1 


31.6 


31.6 





30.8 


30.9 


—.1 


2000 


28.1 


28.1 





28.7 


28.6 


.1 


28.0 


28.1 


—.1 


2500 


25.6 


25.5 


.1 


26.1 


26.1 





25.5 


25.7 


—.2 


3000 


23.4 


23.5 


—.1 


23.8 


23.9 


—.1 


23.3 


23.5 


.2 


3500 


21.4 


21.5 


—.1 


21.8 


21.7 


.1 


21.3 


21.5 


— 2 


4000 


19.6 


19.4 


.2 


20.0 


20.0 





19.5 


19.6 


—.1 





41.5 


41.4 


.1 


41.4 


41.2 


.2 


41.6 


41.4 


.2 


500 


37.3 


37.2 


.1 


37.2 


37.3 


—.1 


37.4 


37.4 





1000 


33.6 


33.5 


.1 


33.6 


33.6 





33.7 


33.8 


—.1 


1500 


30.5 


30.5 





30.4 


30.4 





30.5 


30.6 


—.1 


2000 


27.7 


27.9 


—.2 


27.6 


27.6 





27.7 


27.9 


—.2 


2500 


25.2 


25.3 


—.1 


25.2 


25.2 





25.3 


25.5 


—.2 


3000 


23.0 


23.1 


—.1 


23.0 


23.1 


—.1 


23.1 


23.2 


—.1 


3500 


21.1 


21.3 


—.2 


21.0 


21.2 


—.2 


21.1 


21.3 


—.2 


4000 


19.3 


19.4 


—.1 


19.3 


19.1 


.2 


19.5 


19.6 


—.1 





41.4 


41.1 


.3 


40.5 


40.3 


.2 


39.3 


39.4 


—.1 


500 


37.2 


37.2 





36.4 


36.5 


—.1 


35.4 


35.3 


.1 


1000 


33.6 


33.6 





32.9 


33.0 


—.1 


32.0 


32.0 





1500 


30.4 


30.5 


—.1 


29.8 


29.8 





29.0 


29.0 





2000 


27.6 


27.7 


—.1 


27.1 


27.0 


.1 


26.4 


26.4 





2500 


25.2 


25.2 





24.7 


24.6 


.1 


24.1 


24.1 





3000 


23.0 


23.1 


— 1 


22.6 


22.5 


.1 


22.0 


22.1 


—.1 


3500 


21.0 


21.2 


2 


20.7 


20.7 





20.2 


20.3 


—.1 


4000 


19.3 


19.5 


— 2 


19.0 


19.0 





18.5 


18.5 








38.0 


38.3 


— 3 


39.6 


39.5 


.1 


42.0 


41.8 


.2 


500 


34.1 


33.5 


.6 


35.6 


35.6 





37.7 


37.9 


—.2 


1000 


30.9 


30.4 


.5 


32.2 


32.1 


.1 


34.0 


34.2 


—.2 


1500 


28.0 


27.8 


.2 


29.2 


29.3 


— 1 


30.8 


31.0 


—.2 


2000 


25.5 


25.3 


.2 


26.6 


215.7 


— 1 


28.0 


28.2 


— 2 


2500 


23.2 


23.0 


.2 


24.2 


24.3 


— 1 


25.5 


25.6 


— 1 


3000 


21.3 


21.3 





22.2 


22.3 


— 1 


23.3 


23.4 


— 1 


3500 


19.6 


19.6 





20.3 


20.4 


— 1 


21.3 


21.5 


—.2 


4000 


18.0 


17.8 


.2 


18.6 


18.7 


— 1 


19.5 


19.6 


—.1 





42.1 


42.0 


.1 


41.7 


41.7 





40.6 


40.6 





500 


37.8 


37.9 


—.1 


37.4 


37.4 





36.5 


36.6 


— 1 


1000 


34.1 


34.1 





33.8 


33.9 


—.1 


33.0 


33.0 





1500 


30.9 


30.9 





30.6 


30.6 





29.9 


30.0 


— 1 


2000 


28.0 


28.1 


—.1 


27.8 


27.9 


—.1 


27.2 


27.2 





2500 


25.5 


25.7 


—.2 


25.3 


25.5 


—.2 


24.8 


24,8 





3000 


23.3 


23.3 





23.1 


23.2 


—.1 


22.6 


22.7 


—.1 


3500 


21.3 


21.3 





21.2 


21.4 


—.2 


20.7 


20.8 


—.1 


4000 


19.5 


19.5 





19.4 


19.6 


,2 


19.0 


18.9 


.1 



307 



6. Experiments with the Full Cylinder and Short Suspension. 



m 


Computed 
An 


Observed 
■Am 


C—O 


Computed 
An 


Observed 
An 


C—O 


Computed 
An 


Observed 
An 


C—O 





12.4 


12.45 





12.1 


12.15 





13.2 


13.15 





730 


11.7 


11.6 


.1 


11.4 


11.35 





12.5 


12.5 





14 GO 


11.0 


11.0 





10.8 


10.65 


.1 


11.8 


11.95 


—.1 


2190 


10.4 


10.4 





10.2 


10.25 


—.1 


11.2 


11.1 


.1 


2920 


9.9 


9.8 


.1 


9.6 


9.55 


.1 


10.5 


10.55 


—.1 


3G50 


9.3 


9.3 





9.1 


9.1 





9.9 


10.0 


—.1 


4380 


8.8 


8.85 





8.6 


8.6 





9.4 


9.45 


—.1 


5110 


8.3 


8.35 





8.1 


8.15 





8.9 


8.9 





5840 


7.9 


7.85 





7.7 


7.65 





8.4 


8.5 


—.1 





13.0 


13.0 





12.1 


12.05 





12.6 


12.55 





730 


12.3 


12.25 





11.5 


11.7 


—.2 


12.0 


12.0 





1460 


11.6 


11.55 





10.9 


11.05 


—.1 


11.3 


11.3 





2190 


10.9 


11.05 


— 1 


10.4 


10.5 


— 1 


10.8 


10.75 





2920 


10.3 


10.3 


■ 


9.8 


9.95 


— 1 


10.2 


10.1 


.1 


3650 


9.8 


9.75 





9.5 


9.45 


.1 


9.7 


9.65 


.1 


4380 


9.2 


9.2 





8.9 


9.05 


— 1 


9.2 


9.25 





5110 


8.7 


8.8 


—.1 


8.5 


8.6 


—.1 


8.8 


8.8 





5840 


8.2 


8.2 





8.0 


8.05 





8.4 


8.35 








12.4 


12.4 





12.4 


12.4 





13.5 


13.5 





730 


11.8 


11.65 


.1 


11.8 


11.75 





12.8 


12.75 


.1 


1460 


11.2 


11.15 


.1 


11.2 


11.2 





12.2 


12.35 


—.1 


2190 


10.6 


10.55 


.1 


10.6 


10.6 





11.6 


11.65 





2920 


10.1 


10.2 


— 1 


10.1 


10.05 


.1 


11.0 


11.0 





3650 


9.6 


9.6 





9.6 


9.55 


.1 


10.5 


10.55 





4380 


9.2 


9.15 





9.2 


9.15 


.1 


10.0 


10.0 





5110 


8.7 


8.7 





8.7 


8.65 


.1 


9.5 


9.55 





5840 


8.3 


8.35 





8.4 


8.3 


.1 


9.1 


9.1 








13.0 


12.95 





13.6 


13.6 


"o 


13.9 


14.0 


—.1 


730 


12.4 


12.4 





12.9 


13.0 


—.1 


13.2 


13.2 





1460 


11.7 


11.85 


— 1 


12.3 . 


12.05 


.2 


12.5 


12.65 


—.1 


2190 


11.2 


11.3 


— 1 


11.6 


11.5 


.1 


11.9 


11.95 





2920 


10.6 


10.8 


— 2 


11.1 


11.0 


.1 


11.3 


11.4 


—.1 


3650 


10.1 


10.1 





10.5 


10.55 





10.7 


10.55 


.2 


4380 


9.7 


9.7 





10.0 


10.0 





10.2 


10.25 





5110 


9.2 


9.2 


o 


9.5 


9.55 





9.7 


9.8 


—.1 


5840 


8.8 


8.9 


-1 1 


9.0 1 


9.0 


o 1 


9.3 


9.2 


.1 



— 308 



7. Experiments with the Empty Cylinder and Long Suspension. 



m 


Computed 
A. 


Observed 
-Am 


O—O 


Computed 

A m 


Observed 
■A-m 


O—O 


Computed 


Observed 
■A m 


0—0 





37.5 


37.7 


—.2 


38.2 


37.8 


A 


37.6 


37.6 





500 


28.1 


27.7 


.4 


28.6 


28.8 


—.2 


28.2 


28.3 


—.1 


1000 


21.4 


21.0 


.4 


21.8 


22.0 


—.2 


21.5 


21.4 


.1 


1500 


16.5 


16.6 


—.1 


16.8 


16.7 


.1 


16.6 


16.7 


—.1 


2000 


12.9 


13.1 


—.2 


13.1 


13.0 


.1 


12.9 


13.0 


—.1 





38.0 


37.9 


.1 


39.2 


38.8 


.4 


38.9 


38.8 


.1 


500 


28.4 


28.6 


—.2 


29.3 


29.4 


—.1 


29.1 


29.2 


—.1 


1000 


21.7 


21.6 


.1 


22.3 


22.4 


—.1 


22.1 


22.3 


—.2 


1500 


16.7 


16.8 


—.1 


17.2 


17.2 





17.1 


17.2 


—.1 


2000 


13.0 


13.0 





13.4 


13.4 





13.3 


13.2 


.1 





40.3 


40.4 


—.1 


40.8 


40.8 





38.0 


38.0 





500 


30.0 


29.9 


.1 


30.4 


30.4 





28.4 


28.4 





1000 


22.8 


22.5 


.3 


23.1 


23.1 





21.7 


21.7 





1500 


17.6 


17.8 


—.2 


17.7 


17.9 


—.2 


16.7 


16.8 


—.1 


2000 


13.7 


13.9 


—.2 


13.8 


13.9 


—.1 


13.0 


13.0 








37.9 


37.9 





40.2 


40.2 





39.9 


40.0 


—.1 


500 


28.4 


28.4 





30.0 


30.0 





29.7 


29.6 


.1 


1000 


21.6 


21.5 


.1 


22.8 


22.9 


—.1 


22.6 


22.5 


.1 


1500 


16.7 


16.8 


—.1 


17.5 


17.6 


—.1 


17.4 


17.3 


.1 


2000 


13.0 


13.0 





13.6 


13.9 


—.3 


13.6 


13.6 








39.8 


40.0 


— 2 


39.2 


39.1 


.1 


40.7 


40.5 


.2 


500 


29.7 


29.7 





29.3 


29.6 


—.3 


30.3 


30.4 


—.1 


1000 


22.6 


22.4 


.2 


22.3 


22.3 





23.0 


23.1 


—.1 


2500 


17.4 


16.7 


.7 


17.2 


17.2 





17.7 


17.6 


.1 


2000 


13.5 


13.2 


.3 


13.4 


13.4 





13.8 


13.7 


.1 





40.4 


40.4 





40.1 


40.2 


—.1 


40.4 


40.4 





500 


30.1 


30.3 


—.2 


29.9 


29.9 





30.1 


30.2 


—.1 


1000 


22.9 


22.7 


.2 


22.7 


22.7 





22.9 


22.7 


.2 


1500 


17.6 


17.5 


.1 


17.5 


17.5 





17.6 


17.6 





2000 


13.7 


13.7 





13.6 


13.6 





13.7 


13.6 


.1 





38.9 


38.9 





38.6 


38.8 


— 2 


38.7 


38.7 





500 


29.1 


29.1 





28.8 


28.6 


.2 


28.9 


28.8 


.1 


1000 


22.1 


21.9 


.2 


22.0 


21.9 


.1 


22.0 


22.2 


—.2 


1500 


17.0 


17.0 





16.9 


16.9 





17.0 


17.1 


—.1 


2000 


13.3 


13.4 


— 1 


13.2 


13.2 





13.2 


13.4 


—.2 





39.5 


39.5 





38.1 


38.0 


.1 


38.5 


38.5 





500 


29.5 


29.8 


—.3 


28.5 


28.6 


—.1 


28.8 


29.0 


— 2 


1000 


22.4 


22.4 





21.7 


21.8 


—.1 


21.9 


21.9 





1500 


17.3 


17.2 


.1 


16.8 


16.6 


.2 


16.9 


16.8 


.1 


2000 


13.4 


13.4 





13.0 


12.9 


.1 


13.2 


13.0 


.2 



— 309 



7. Experiments with the Empty Cylinder and Long Suspension. — Continued. 



m 


Computed 


Observed 
An 


C—O 


Computed 
An 


Observed 
An 


c—o\ 


Computed 
An 


Observed 


C—O 





39.2 


38.8 


.4 


38.5 


38.4 


.1 


39.0 


39.0 





500 


29.3 


29.5 


—.2 


28.8 


29.0 


— .2 


29.1 


29.2 


—.1 


1000 


22.3 


22.4 


—.1 


21.9 


21.9 





22.2 


22.2 





1500 


17.2 


17.2 





16.9 


16.9 





17.1 


17.2 


—.1 


2000 


13.4 


13.6 


—.2 


13.2 


18.2 





13.3 


13.4 


—.1 





39.9 


39.9 





38.4 


38.4 





38.9 


38.9 





500 


29.8 


29.8 





28.7 


28.8 


1 


29.1 


29.2 


—.1 


1000 


22.G 


22.8 


—.2 


21.9 


22.0 


1 


22.1 


22.1 





1500 


17.4 


17.6 


— .2 


16.9 


17.0 


1 


17.1 


17.1 





2000 


13.6 


13.6 





13.1 


13.1 





13.3 


13.2 


.1 





38.6 


38.6 





39.1 


39.1 





36.8 


36.8 





500 


28.9 


28.9 





29.2 


29.3 


1 


27.6 


27.6 





1000 


22.0 


21.9 


.1 


22.2 


22.1 


.1 


21.1 


21.1 





1500 


16.9 


16.8 


.1 


17.1 


17.0 


.1 


16.3 


16.3 





2000 


13.2 


13.1 


.1 


13.3 


13.1 


.2 


12.6 


12.6 








36.3 


36.3 





38.3 


38.3 





39.4 


39.3 


.1 


500 


27.3 


27.4 


1 


28.6 


28.7 


1 


29.4 


29.5 


—.1 


1000 


20.8 


21.0 


2 


21.8 


21.8 





22.4 


22.4 





1500 


16.1 


16.1 





16.8 


16.8 





17.2 


17.3 


—.1 


2000 


12.5 


12.5 





13.1 


13.1 





13.4 


13.5 


—.1 





40.1 


40.1 





39.4 


39.3 


.1 


38.8 


38.8 





500 


29.9 


29.9 





29.4 


29.5 


— 1 


29.0 


29.4 


—.4 


1000 


22.7 


22.6 


.1 


22.4 


22.4 





22.1 


22.0 


.1 


1500 


17.5 


17.4 


.1 


17.2 


17.3 


— 1 


17.0 


17.0 





2000 


13.6 


13.5 


.1 


13.4 


13.5 


— 1 


13.2 


13.3 


—.1 





40.0 


40.0 





38.7 


38.7 





38.8 


38.9 


—.1 


500 


29.8 


30.0 


—.2 


28.9 


28.9 





29.0 


28.9 


.1 


1000 


22.7 


22.5 


.2 


22.0 


21.9 


.1 


22.1 


22.0 


.1 


1500 


17.4 


17.3 


.1 


17.0 


16.9 


.1 


17.0 


17.1 


—.1 


2000 


13.6 


13.6 





13.2 


13.2 





13.2 


13.3 


—.1 





39.1 


39.1 





38.5 


38.5 





38.7 


38.7 





500 


29.2 


29.1 


.1 


28.8 


28.8 





28.9 


29.0 


— 1 


1000 


22.2 


22.1 


.1 


21.9 


21.8 


.1 


22.0 


22.0 





1500 


17.1 


17.1 





16.9 


16.9 





17.0 


16.9 


.1 


2000 


13.3 


13.3 





13.2 


13.1 


.1 


13.2 


13.2 








37.7 


37.7 





38.8 


38.8 





38.1 


38.2 


—.1 


500 


28.2 


28.0 


.2- 


29.0 


29.0 





28.5 


28.5 





1000 


21.5 


21.5 





22.1 


22.0 


.1 


21.7 


21.5 


.2 


1500 


16.6 


16.7 


—.1 


17.0 


16.9 


.1 


16.8 


16.6 


.2 


2000 


12.9 


12.9 





13.2 


13.1 


•1 


13.0 


12.9 


.1 



310 — 





8. Experiments 


with tlie 


Empty Cylinder and the Short Suspension. 




m 


Computed 
An 


Observed 
An 


C—O 


Computed 


Observed 

An 


C—O 


Oomputed 
An 


Observed 
An 


C—O 





11.4 


11.4 





12.2 


12.3 


— 1 


13.3 


13.3 





800 


9.5 


9.4 


,i 


10.1 


10.15 





11.0 


10.95 





1600 


7.9 


7.7 


2 


8.5 


8.45 





9.1 


9.05 





2400 


6.6 


6.5 


1 


7.1 


.7.05 





7.6 


7.55 





3200 


5.5 


5.5 


.0 


6.0 


5.9 


.1 


6.3 


6.3 





4000 


4.7 


4.6 


.1 


5.0 


4.9 


.1 


5.3 


5.3 





4800 


4.0 


3.9 


.1 


4.2 


4.1 


.1 


4.4 


4.5 


—.1 





13.3 


13.3 





13.4 


13.4 





12.1 


12.1 





800 


11.0 


11.1 


—.1 


11.3 


11.3 





10.2 


10.25 





1G00 


9.1 


9.15 





9.5 


9.4 


.1 


8.7 


8.65 





2400 


7.6 


7.8 


—.2 


8.0 


8.05 





7.3 


7.4 


— 1 


3200 


6.3 


6.2 


.1 


6.8 


6.7 


.1 


6.2 


6.25 





4000 


5.3 


5.4 


—.1 


5.8 


5.7 


.1 


5.3 


5.3 





4800 


4.4 


4.45 





4.9 


4.9 





4.5 


4.5 








12.3 


12.2 


.1 


13.0 


13.0 





13.2 


13.15 





800 


10.5 


10.5 





11.0 


11.0 





11.2 


11.15 





1G00 


8.9 


9.0 


—.1 


' 9.4 


9.3 


.1 


9.5 


9.6 


— 1 


2400 


7.6 


7.85 


.2 


8.0 


8.0 





8.1 


8.0 


.1 


3200 


6.5 


6.7 


—.2 


6.9 


6.9 





6.9 


6.85 





4000 


5.6 


5.75 


—.2 


5.9 


5.9 





5.9 


5.95 





4800 


4.8 


4.95 


—.1 


5.0 


5.1 


—.1 


5.1 


5.1 








13.0 


12.95 





14.1 


14.25 


—.1 


12.9 


12.9 





800 


11.0 


1I\05 





11.9 


11.9 





10.9 


10.95 





1600 


9.4 


9.4 





10.1 


10.1 





9.3 


9.15 


.1 


2400 


8.0 


8.05 


—.1 


8.6 


8.05 


—.1 


7.9 


7.85 





3200 


6.8 


6.9 


—.1 


7.3 


7.35 


—.1 


6.7 


6.85 


—.1 


4000 


5.8 


5.85 





6.2 


5.95 


.3 


5.7 


5.85 


—.1 


4800 


5.0 


5.0 





5.3 


5.25 


.1 


4.9 


4.9 






In the computation of these values, there has been no regard 
to the resistance arising from the wires of suspension. The dif- 
ference between the values of H 2 may be attributed to the uncer- 
tainty of the observations, and those of H x may, perhaps, be ac- 
counted for, in the same way. The value of ff 2 is nearly ten 
times as great as that which is given by the observations of Borda 
upon the resistance of the atmosphere. It must, therefore, be 
doubtful, whether the observed diminution of the arcs of vibration 



— 311 — 

of the pendulum is, wholly or principally, due to the medium in 
which it vibrates, or to some more latent cause. This doubt is 
much increased by the discussion of the observations of Baily. 

534. In Baily's experiments, various pendulums, which were 
mostly spheres and cylinders, were vibrated in the receiver of an 
air-pump, with the air either at its ordinary pressure, or at the 
small density of about one thirtieth of an atmosphere. For the 
full and exact description of the pendulums the original memoir 
must be consulted, but the following brief description is sufficient 
for the present purpose. Numbers 1, 2, 3, and 4 are spheres of 
platina, lead, brass, and ivory, all of the same diameter, which is 
somewhat less than 1£ inches, and of which the weights with their 
vibrating appendages are, respectively 9050, 4648, 3217, and 776 £ 
grains. Nos. 5, 6, and 7 are spheres of lead, brass, and ivory, all 
of the same diameter, which is 2.06 inches, and of which the 
weights are respectively, 13019, 9302, and 2066 i grains. Nos. 8 
and 9 are the same spheres of lead and ivory with those of Nos. 5 
and 7, but suspended from a wire passing over a small cylinder 
instead of from a knife edge. In Nos. 10, 11, 12, and 13 the 
vibrating mass was a brass cylinder, of which the diameter of the 
base is 2.06 inches, the altitude 2.06 inches, and the weight 14190 
grains ; in Nos. 10 and 13 the axis of the cylinder coincides with 
that of the pendulum rod, but the rod of No. 13, which was also 
adopted in Nos. 11 and 12, was a thick brass wire 0.185 inch in 
diameter, 371 inches long, and weighing 2050 grains ; in Nos. 11 
and 12 the axis of the cylinder was horizontal, in No. 11 it was 
perpendicular to the plane of vibration, and in No. 12 it was in the 
plane of vibration. No. 14 is a cylinder of lead, of which the 
diameter of the base is 2.06 inches, the altitude 4 inches, the weight 
34500 grains, and the axis coincident with the rod of the pendulum. 
In Nos. 15, 16, 17, 18, and 19 the vibrating mass was a hollow cyl- 



— 312 — 

incler of the same position and external dimensions with No. 14 ; 
in No. 15 both ends were open; in No. 16 the top was open and 
the bottom closed; in No. 17 the top was closed and the bottom 
open; in No. 18 both ends were closed; in No. 19 an inner sliding 
tube was removed so as to reduce the weight; and the weights, 
with the inclosed air, were, respectively, 8497, 8922, 8622, 9048, 
and 7250 grains. No. 20 is a lens of lead 2.06 inches in diameter, 
an inch thick in the middle, with a flat circumference of about a 
quarter of an inch wide, and a weight of 6505 grains. No. 21 is a 
solid copper cylindrical rod of 0.41 inch in diameter, 58.8 inches 
long, and weighing 16810 grains. In Nos. 25, 26, 27, 28, 29, 30, 31, 
32, 33, and 34, the vibrating masses were convertible pendulums, 
formed of plane bars, and they are vibrated successively with each 
of their points of suspension, which were knife edges; in Nos. 25 
and 26 the bar was brass, two inches wide, three eighths of an inch 
thick, 62.2 inches long, and weighing 121406 grains; in Nos. 27 
and 28 it was copper of the same width with the brass bar, half 
an inch thick, 6'2.5 inches long, and weighed 155750 grains ; in 
Nos. 29 and 30, it was iron of the same width and thickness with the 
copper bar, 62.1 inches long, and weighed 140547 grains ; in Nos. 31, 
32, 33, and 34 it was a doubly convertible brass bar, three quarters of 
an inch thick, 62 inches long, and weighed 231437 grains. In Nos. 35, 
36, 37, and 38, a doubly convertible pendulum, made of a brass cylin- 
drical tube of lh inches in diameter, 56 inches long, and weighing 
81047 grains was vibrated upon a knife edge with all four of its 
planes of suspension. No. 39 is a mercurial pendulum. Nos. 40 and 
41 are clock pendulums in which the vibrating mass was a leaden 
cylinder 1.8 inches in diameter, 13.5 inches long, and weighing 
93844 grains ; in No. 40 it was suspended from a spring, by a cylin- 
drical rod of deal of three eighths of an inch in diameter, and in 
No. 41 by a flat rod of deal one inch wide, 0.14 inch thick in the 



— 313 — 

middle of its width and bevelled on each side to a thin edge, which 
was opposed to the direction of its motion. 

In the discussion of Baily's experiments, the value of H % is 
neglected, because it is of small influence, and the arcs of vibration, 
being usually given only for the beginning and end of the experi- 
ment, are just sufficient to determine one of the quantities II X 
and II 2 ; and the values of H^ are not reduced to the same density 
of air. The ratio of the value of H^ for the ordinary state of the 
air to its value in the exhausted receiver, varies from 1.9 to 4.2, in- 
stead of being about 30, which it should be if it were proportional 
to the density of the air ; the value of this ratio in the following 
table is expressed by J.. The total resistance to the motion of the 
pendulum, supposed to be proportional to the velocity is, for the 
unit of velocity, expressed by H" in the table ; and this same re- 
sistance, reduced to the unit of weight, is expressed by H{. 

The observation of the arcs of vibration in Baily's experiments 
is limited to the initial and final arcs, and the direct comparison of 
the computed and observed arcs is, consequently, quite unnecessary, 
and cannot contribute to verify the accuracy of the hypothesis upon 
which the computation is based. The only two cases in which an 
intermediate arc was observed with Nos. 6 and 14 seem to sustain 
the hypothesis ; for they differ from it slightly, but in opposite direc- 
tions. 

The diversity of the values of H x indicates that the resisting 
force of the motion to the pendulum demands a new experimental 
investigation, conducted with a direct object to its determination ; 
and that, until such an investigation has been made, the length of 
the seconds pendulum must be regarded as liable to an unknown 
error. 

40 



— 314 — 



Values of Hi in Bailg's Experiments upon the Vibrations of Pendulums. 



No. of 
Pendulums. 


Barometer. 


H x 


Hi 


H? 


J 


1 


0.7689 


.0673 


.000077 


.000132 


2.68 


3 


0.7646 


.0702 


.000080 


.000384 


2.62 


2 


0.7523 


.0662 


.000075 


.000250 


2.55 


4 


0.7660 


.0561 


.000063 


.001272 


2.71 


6 


0.7638 


.0570 


.000123 


.000204 


2.74 


7 


0.7630 


.0538 


.000116 


.000864 


' 2.62 


5 


0.7644 


.0627 


.000128 


.000] 61 


3.18 


9 


0.7682 


.0589 


.000127 


.000945 


2.86 


8 


0.7677 


.1021 


.000219 


.000261 


2.92 


10 


0.7652 


.0651 


.000179 


.000194 


3.42 


11 


0.7637 


.0558 


.000270 


.000256 


2.62 


12 


0.7623 


.0603 


.000290 


.000277 


3.33 


13 


0.7552 


.0571 


.000235 


.000262 


2.98 


18 


0.7491 


.0535 


.000285 


.000484 


3.27 


15 


0.7554 


.0658 


.000350 


.000635 


4.10 


16 


0.7495 


.0595 


.000292 


.000505 


2.95 


17 


0.7584 


.0558 


.000297 


.000531 


3.39 


14 


0.7747 


.0592 


.000315 


.000140 


4.22 


19 


0.7620 


.0510 


.000272 


.000578 


3.33 


20 


0.7620 


.0656 


.000065 


.000156 


2.09 


21 


0.7575 


.0661 


.000742 


.000682 


2.72 


25 


0.7522 


.0789 


.005606 


.000333 


3.32 


26 


0.7465 


.0756 


.004782 


.000319 


3.74 


31 


0.7522 


.1555 


.003666 


.000245 


3.32 


32 


0.7520 


.1581 


.003673 


.000245 


3.55 


34 


0.7529 


.1661 


.003772 


.000251 


3.72 


33 


0.7535 


.1417 


.003480 


.000232 


3.13 


35 


0.7595 


.0739 


.003091 


.000589 


3.48 


36 


0.7627 


.0660 


.002763 


.000526 


3.31 


37 


0.7577 


.0701 


.002931 


.000558 


3.39 


38 


0.7564 


.0659 


.002760 


.000526 


2.97 


39 


0.7622 




.001396 


.000209 


1.87 


41 


0.7573 


.0664 


.001260 


.000207 


2.52 


40 


0.7589 


.0769 


.001299 


.000213 


2.39 



— 315 — 



Values of Hi in Baily's Experiments upon the Vibrations of Pendulums. — Continued. 



No. of 
Pendulums. 


Barometer. 


H x 


m 


H{' 


J 


1 


0.0288 


.0251 


.000028 


.000049 


2.68 


3 


0.0294 


.0267 


.000031 


.000146 


2.62 


2 


0.0265 


.0259 


.000030 


.000098 


2.55 


4 


0.0347 


.0284 


.000024 


.000470 


2.71 


6 


0.0268 


.0285 


.000044 


.000074 


2.74 


7 


0.0270 


.0282 


.000044 


.000330 


2.62 


5 


0.0290 


.0275 


.000042 


.000050 


3.18 


9 


0.0360 


.0282 


.000044 


.000331 


2.86 


8 


0.0299 


.0348 


.000075 


.000089 


2.92 


10 


0.0239 


.0190 


.000052 


.000057 


3.42 


11 


0.0478 


.0213 


.000103 


.000098 


2.62 


12 


0.0348 


.0182 


.000088 


.000083 


3.33 


13 


0.0370 


.0192 


.000092 


.000089 


2.98 


18 


0.0300 


.0164 


.000087 


.000148 


3.27 


15 


0.0271 


.0164 


.000097 


.000148 


4.10 


16 


0.0266 


.0186 


.000099 


.000171 


2.95 


17 


0.0362 


.0165 


.000088 


.000157 


3.39 


14 


0.0298 


.0139 


.000074 


.000033 


4.22 


19 


0.0305 


.0154 


.000083 


.000174 


3.33 


20 


0.0305 


.0313 


.000031 


.000074 


2.09 


21 


0.0288 


.0244 


.000274 


.000251 


2.72 


25 


0.0313 


.0238 


.001505 


.000101 


3.32 


26 


0.0325 


.0202 


.001277 


.000086 


3.74 


31 


0.0414 


.0469 


.001105 


.000074 


3.32 


32 


0.0391 


.0439 


.001034 


.000069 


3.55 


34 


0.0410 


.0431 


.001014 


.000067 


3.72 


33 


0.0463 


.0472 


.001111 


.000074 


3.13 


35 


0.0384 


.0213 


.000888 


.000170 


3.48 


36 


0.0367 


.0200 


.000834 


.000160 


3.31 


37 


0.0422 


.0206 


.000859 


.000166 


3.39 


38 


0.0412 


.0222 


.000930 


.000178 


2.97 


39 


0.0477 




.000747 


.000112 


1.87 


41 


0.0457 


.0263 


.000498 


.000083 


2.52 


40 


0.0434 


.0320 


.000543 


.000089 


2.39 



— 316 



THE TAUTOCHRONE. 



535. The consideration of the pendulum leads, directly, to 
the investigation of that curve, upon which the duration of the 
vibration is independent of the length of the arc of oscillation. 
Such a curve is called a tautochrone, and is readily determined when 
the body is only subject to the action of fixed forces. 

536. If the force which acts in the direction of the motion 
of the body is denoted by S, the equation of its motion is 

In the case in which JS is a function of s, let s denote the 
point, at which the velocity vanishes, or the extremity of the arc 
of vibration. Hence 






and if the origin of coordinates is at the point of maximum velocity, 
the time of vibration is determined by the equation 

Tz=z ( y/2 

o 

If h—-, 

v 

if £2 is a function of s expressed by £2 S , and if s is written instead 
of s Q , the value of T becomes 

T— t s ^ 2 

M —Jkij(si A —a,y 



In order that the special value of the arc may disappear from 



— 317 — 

this integral, it is obvious that S2 S has the form 

S2 S = A — Bs 2 , 
which reduces the value of T to 

JksJBsl(l — K 2 ) \ 2B' 
The tangential force along the curve is, therefore, 
S—DJl = — 2Bs. 

537. If F denotes the actual force, which acts upon the body 
in the direction of/, the preceding equation gives for an equation of 
the tantochrone 

Fcos{ = — 2Bs=FB s f, 
or 

A—Bs 2 =fF. 

538. In the case in which the body is restricted to move upon a 
curve ivhich rotates uniformly about a fixed axis, the equations and 
notation of §468 combined with the previous section, give for 
the equation of the tantochrone 

A — Bs 2 = ±a 2 u 2 , 

which may assume the form 

s 2 . u? , 

in which a and b are constants. 

539. When the revolving curve is a plane curve, and situated in 
the same plane with the axis of revolution, the notation 



b = a cot i 



a sin 6 = a sin cp sm«, 



— 318 — 

and that of elliptic functions give 

u = b cos 6 , 

s = -^—. ^(p — b cos i^i(f; 



and if r is the inclination of the curve to the axis of rotation, its 
value is 

sin t = — cot i tan £ . 
The maximum of u is b, but its least value, corresponding to 

or 6 = ip i, 

is u = b cos i ; 

and the corresponding value of s is 

s = J^asmi. 

The curve consists of several branches, which form cusps by their 
mutual contact at their extremities, and it resembles the cycloid in its general 
character. 

540. In the case of a heavy body moving upon a plane vertical curve, 
let v denote the angle which the radius of curvature () makes with 
its horizontal projection, and the equation (317u) gives 

F 
s = — ^cosv, 

F . 

which is the equation of the cycloid referred to its radius of curva- 
ture and angle of direction, so that the cycloid is the tautochrone of a 
free heavy body in a vacuum. The same curve, drawn upon the de- 



— 319 — 

veloped surface, is the taatochrone of a heavy body, moving upon a vertical 
cylinder. 

541. Every curve may be regarded as being upon the surface 
of its vertical cylinder of projection ; and, therefore, the tautochrone 
of a heavy body moving in a vacuum upon any surface ivhatever, is the 
intersection of the surface tvith such a vertical cylinder, that the intersection 
is a cycloid upon the developed vertical cylinder. The determination of 
the tautochrone upon any surface is thus reduced to a problem of 
pure geometry. If the axis of z is the upward vertical, and if z is 
the height of the lowest point of the curve above the origin, the 
equation (317i 6 ) becomes, in the present case, 

542. If a heavy body is restricted to move upon a cylinder of which 
the axis is horizontal, and of which the equation of the base is 

Q 1 = na cos 7^ sin" -1 ^, 

in which v x is the angle, which the radius of curvature, denoted by 
q 1} makes with the upward vertical ; and when the cylinder is devel- 
oped into a vertical plane, if y is the height of the moving body 
above the horizontal line, which corresponds to the lowest side of 
the undeveloped cylinder, the value of y is 

y ■=. a sin n v x . 

The force of gravity, resolved in a direction tangential to the 
cylinder, is 

gsmv 1 =g^; 

so that the present problem corresponds to that of a body moving in a ver- 



— 320 — 

tical plane, and subject to a force which is fixed in direction, and propor- 
tional to some poiver of the height above a given level. The equation 
(319 13 ) gives for the equation of the tautochrone 



B 



»+i* — na Mfl+i) 



- s 2 4- z = j (sin v x D v y) = — ^— a sin" +1 v, = -^— \ y -\ 

543. If r denotes the angle which the radius of curvature (q) 
of the tautochrone makes with the upward vertical in the developed 
cylinder, the equation (317 14 ) gives 

2B 

sin v smv,=- — s, 
9 

which, substituted in (320 5 ), reduces the equation of the tautochrone to 

B o . n /25SV+1 

- sf-\- z = — j— a I —r— ) 

g ' n -\- 1 v? sm y/ 

544. When z vanishes in the problem of the preceding sec- 
tion, the equation of the tautochrone becomes 



, . ttl / nag /yY±± 



n + l z . _«. 

or C = _ -osin"- 1 ^ 



l 



in which 



(«)- , f=&r©" 



so that ^e tautochrone on the developed cylinder of § 542 & of ^e same 
trigonometric class of curves with the base of the cylinder, when it passes 
through the loivest side of the undeveloped cylinder. This case is impos- 
sible, when n is included between positive and negative unity ; for 
when n is negative and, independently of its sign, less than unity, s 
becomes infinite when y vanishes, but when n is positive and less 



— 321 — 
than unity, the derivative of (320 19 ), which is 

D ljS = cosec v = I Jjf\£)~^ r , 

gives the impossible result that cosec v vanishes with y. 

545. The differential equation of the tautochrone, in the case of § 
542, referred to rectangular coordinates upon the developed cylinder, is 
readily obtained from the equations of § 542, which give 

^(lM(I) 1+i -^)(^+i)> 

in which 

7.2 9 " + 1 

/r IB na ' 

and the axis of x is horizontal. 

In the case of § 544, in which s vanishes, this equation becomes 



D^+l=P^-\ 



546. In the case in which n is unity, that is, in ivhich the base 
of the cylinder is a cycloid, the equation of the tautochrone on the developed 
cylinder, becomes 

When s vanishes, this curve is reduced to a straight line, but 
in all other cases, its form, if it is infinitely extended in the plane 
of the developed cylinder, resembles the hyperbola. By the adop- 
tion of the notation 

. „ . 2aB 

sir I = , 

9 

y= y/(2as' )sec9, 
41 



— 322 — 

and that of elliptic functions, its equation may be expressed in the 
forms 



sj 9 -^ (cos 6 tan 9 + % y — 8 4 9) . 



547. .7^ « /^a^ £o<^/ /s restricted to move upon a surface of revo- 
lution about a vertical axis, of which the equation of the meridian 
curve is that of (319 17 ). If y is the distance of the body on the 
meridian curve from the lowest point of the surface, the value of 
y is given by the equation (319 25 ), and the force of gravity, resolved 
in a direction tangential to the meridian curve is expressed by 
(319 29 ), so that the present problem resembles that of a body 
moving in a plane, and subject to a force, which is directed 
towards a fixed point in the plane, and is proportional to some 
power of the distance from that point. The equation (317k) of the 
tautochrone, gives 

7? ,2 9 y — 2/0 

in which m is the reciprocal of n, and y the value of y at the lowest 
point of the tautochrone. 

548. When m vanishes, the surface of revolution is a right 
cone, and the equation (322 ]9 ) becomes 

Bs 2 = g(y—y ). 

By means of the notation 

sin 2 4 = y(>^- y ), 

sec p = 1 -j -; 



— 323 — 

the angle (ep) which y makes with y in the developed cone is given 
by the formula 

tan 1(6 + h<p) tan } 0] = ^^ ; 

so that ^e ^ofor equation of this tautochrone upon the developed cone is 
expressed by the combination of (322 28 ) and (323 3 ). 

549. When y vanishes, ft also vanishes, and the equation 
(323 3 ) becomes 

fl + *9> + cotfl=0. 

550. When m is unity, the surface of revolution is cycloidal 
and the equation (322 J9 ), becomes 

aBs 2 = ig(y 2 —y 2 ), 

which becomes the meridian curve itself, when y vanishes. 

551. In the case given in (322 u ), of a body moving in a plane 
and subject to a force, ivhich is directed towards a fixed point in the plane, 
and is proportional to some poiver (in) of the distance from that point, 
the equation of the tautochrone may be given in the form 

s 2 = A(r m+1 — r m+1 ), 

in which the attracting point is the origin of polar coordinates. 
The polar differential equation is 

552. If the attraction or repulsion of the point had been any function 
whatever of the distance from the origin, the equation of the tautochrone 
would have assumed the form 

s z=Fr — Fr , 



— 324 — 

in which F denotes the function of which the derivative expresses 
the given law of attraction. This equation may therefore assume 
the form 

x 2 -4- y 2 = r 2 = Si, 

in which S ± is a function of JS. If then v is the angle which the 
radius of curvature makes with the axis of x, the derivatives of 
this equation are 

2 x sin v — 1y cos v = JS[, 

(2#cosv -J- 2y sinv) D s v = 8" — 2 ; 



whence 



2xD s v = S[ sin vD,v-\- (S^ — 2) cos v, 
2yD s v= — 8[ cos v D s v -f {S" — 2) sin y, 
4 8 1 D s v 2 = S? D s v 2 -f ( tf" — 2) 2 , 

,2 

„ 2 n c.2 ^ ! ' Q' 

V X/ i; i Tgtf 2) 2 2 ' 

which is the equation of the tautochrone expressed in terms of the radius 
of curvature and the arc. 

553. The polar differential equation of the tautochrone in the case 
of the preceding section is 

r u r y -f-i — Fr _ Fro , 

which is the same equation with that which is given by Puiseux. 

554. The derivative of (324 16 ) relatively to v is 

2B v q = K, 

so that the elimination of s between (324 16 ) and (324 27 ) gives the 
differential equation of this tautochrone in terms of the radius of curvature 
and the angle of its direction. 



— 325 — 

555. In the case of § 552, when 

$, = a s -J- b } 
the value of S 2 is 

JS 1 = as-\-b — 4 a 2 . 

The equation (324 27 ) becomes therefore 

and the equation of the tautochrone is 

g= \ av , 

which is that of the involute of the circle. This case corresponds to 
that in which the law of the central force is of the form. 

Br{f — t*). 

556. In the case of § 552, when 

% = a (s + bf, 



the value of JS 2 is 








S 2 - 


= f=.m 2 {s-\-bY : 


in which 




9 a 
1 — a 7 



so that a must be positive and less than unity. The equation of 
the tautochrone is, then, 

Q = E e mv , 

ivhich is that of the logarithmic spiral. This case corresponds to that 
in which the law of the central attraction is of the form 

r — r n 



— 326 — 

that is, in which the force is proportional to the distance of the body from 
the circumference of the circle described from the origin as the centre with 
a radius equal to that of the initial position of the body. This case is 
discussed by Puiseux. 

557. In the case of § 552, ivhen the force is proportional to 
the distance from the, origin. The equation (323 31 ) assumes the 
form 



12 



which, with the value of m in (325 22 ), reduces S x and iS 2 to 



r 2 
0' 



#! = as 2, -J- 
The equation of the tautochrone is, therefore, 

of which the integral is 

o — , - Qos(mv) 

^ 1 — a v ' 

in which the arbitrary constant is determined so that v may vanish 
with s. 

The second derivative of this equation gives, for the radius of 
curvature of the second evolute of the tautochrone 

q = m* q 

so that the second evolute is similar to the tautochrone itself. 

In the case in which m is real, which corresponds to that in 



— 327 — 

which a is positive and less than unit}*, this curve runs off to 
infinity in each direction, with a constantly increasing radius of 
curvature. 

In the case in which m is imaginary, the substitution of 

9. 9 

— n= in , 

reduces the equation of the tautochrone to the form 

? = r 5_cos(nv), 

which is the equation of an epicycloid. The epicycloid is formed by the 
external rotation of one circle upon another, when n is less than unity, in 
which case a is negative and the force is repulsive ; but the epicycloid is 
formed by internal rotation, when n is greater than unity, which corresponds 
to the case when a is positive and greater than unity. In either of these 
cases, the initial velocity must not be more than sufficient to carry 
the body to either of the cusps. 

In the case in ivhich a is infinite, the tautochrone is reduced to a 
straight line. 

The example of this section is discussed by Puiseux. 

558. The example of the preceding section embraces the case 
of any force, which is a function of a distance from the origin, in the 
immediate vicinity of the point of greatest velocity. The form of 
the tautochrone, near the point of greatest velocity, in the example of 
\ 552, is typified, therefore, by the epicycloid, or by the curve of equa- 
tion (326 2 i). 

559. The investigation of the tautochrone in a resisting 
medium is postponed to the general case of the chronic curves. 



!28 — 



THE BRACHYSTOCHRONE. 



560. The curve upon which a body moves in the least pos- 
sible time from one given point to another, is called the hrachys- 
tochrone. 

561. The investigation of the general case of a brachysto- 
chrone which is confined to any surface or limited by any condition, 
may be conducted by means of rectangular coordinates. The time 
of transit from the first to the last of the given points may be ex- 
pressed by the equation 



Jx V 



which is to be a minimum. This condition gives, for each of the 
other axes, the equation 



D„(^)-D,D r (S-l) = 0. 



562. When the body is only subject to the action of fixed 
forces, v does not involve either y' or z ', and the preceding equation 
becomes 



D„v 



£ + 2>.(=r)=o, 

or by (316 17 ), 

2>,i2 + t^D.(^")=0. 

563. If the plane of xy is assumed, at each instant, to be 
that in which the body moves, and if the axis of y is taken normal 



— 329 — 

to the path of the body, the preceding equation becomes, if q ex- 
presses the radius of curvature of the path 

Q J 

so that the centrifugal force of the bod// is equal to the normal pressure, 
and the ivhole pressure upon ihe brack// stochrone is double the centrifugal 
force. This proposition was discovered by Euler. 

564. When the normal pressure vanishes, the radius of curvature 
is infinite, which corresponds in general to a point of contrary flexure. 
When there is no force acting upon the body throughout its path, the 
brachystochrone is reduced to a straight line. 

565. Any conditions to which the path must be subject, 
whether elementary such as that it is confined to a given sur- 
face, or integral such as that its whole length is given, must be 
combined with the general condition of brachystochronity by the 
usual methods of the calculus of variations. 

566. If the only force ivhich acts upon the body is directed to a given 
point, and if the path is subject to no conditions, let the plane ofxz be 
assumed to be that which passes through the centre of action and 
the initial element of the path. In this case the equation (328 27 ) 
gives 

cosf=0, l = in, 

or the brachystochrone is contained in a plane which passes through the 
centre of action. 

567. The preceding case includes that in which the centre 
of action is removed to an infinite distance, so that, in the case of 
parallel forces, ihe free brachystochrone is contained in a plane, ivhich is 
parallel to the direction of the forces. 

568. When the body is acted upon by no forces, or only by those 
which are normal to its path and do not tend to change its velocity, the 

42 



— 330 — 

equation (328 13 ) shoivs that the br achy stochr one is the shortest line which 
can be drawn under the given conditions. 

569. When the force is directed towards a fixed centre, the 
equation (329 9 ), combined with (316 18 ) gives, if the centre is adopt- 
ed as the origin 

Z>,.Q 2 



Si — »Qo Q sin 



i 



If p is the perpendicular let fall from the origin upon the 
tangent to the curve, this equation becomes 

Z>,.Sl _2D r p__2D r p 



Si — Si r sin j 2 } ' 

of which the integral is 

CD 

which is the equation of the brachystochrone referred to the radius vector 
and the perpendicular from the origin upon the tangent as the coordinates. 
This form is given by Euler. 

570. When the force in the preceding case, is proportional to the 
distance from the origin so that il has the form 



O — 



ar 



the equation (330 14 ) becomes 
of which the derivative gives 



P 
s P\ 



If v is the anode which o makes with the fixed axis, the de- 



1 

rivative of this last equation gives, by means of the preceding- 
equation 

•which becomes 



if 



2 1 — a pi 
a pi 



The integral of this equation is 



ma pi 



■, Sin (mv) 



so that its second evolute is similar to the br achy stochr one itself. 

When m is real, which corresponds to the case of a repulsive 
force, and ap\ less than unity, this br achy stochr one is a spiral which has 
a cusp at the point at which v vanishes. 

When m is imaginary, the substitution of (327 5 ) reduces (331 n ) 
to the real form 

p = 7, sin (n v) 

s n a p\ v ' 

so that in this case, the brachystochrone is an epicycloid which is formed by 
internal rotation ivhen the force is attractive, and by external rotation when 
the force is repulsive. This case is given by Euler. 

571. When the forces are parallel, the equation (329 3 ) gives, if 
the axis of z is supposed to be in the direction of the forces 

= 2"cot*.2>" 



Si — £2 Q$h 



s -*-^ z s ? 



of which the integral is 

12 — i2 = a sin 2 *, 



— 332 — 

in which a is an arbitrary constant, and this is the equation of the 
br achy stochr one referred to the coordinates, ivhich are z and the inclina- 
tion of the curve to the axis of z ; and the equation, referred to o 
and I as coordinates, is obtained by eliminating z between (331 27 ) and 
(331 31 ). 

572. In the case of a constant force, the preceding equation 
assumes the forms 



g{z — Sq) = a sin 

2a 



2z 

5J 



-sin 



so that, in this case, the br achy stochr one is a cycloid. 

573. When the parallel forces are proportional to the distance from 
a given line, which may be adopted for the axis of x, the value of 
12 has the form 

12 = bz 2 ; 
whence the equation of the br achy stochr one is 

r 

a sin % 
? — V (V 4 + ab sin 2 1) ' 

When the force is repulsive, or luhen it is attractive, but 



a 



this curve consists of branches, ivhich are united by cusps, and resemble 
the cycloid in general form ; but token the force is attractive, and 



^<\l- 



a 



this curve consists of branches which are still united by external cusps ; 
but the middle point of each branch is upon the axis of x, and is a point 



33 



Q 



of inflexion, and the interval between two successive points of inflexion, ex- 
pressed by elliptic integrals, is 

v^(-f)W*")-^(*«)]i 

in -which 

, h 

' a 

In the case of the attractive force, and 

i a 

the equation of the br achy stochr one becomes 

9 = g tan z s5 

which consists of tiuo infinite branches joined by an external cusp, and the 
axis of x is an asymptote to each of the branches. 

574. When the body is subjected to move upon a given sur- 
face, the force by which it is retained upon the surface is perpen- 
dicular to its path, and must be united with the second member of 
equation (329 3 ). Hence it follows that the centrifugal force of the 
body in the direction of the tangent plane to the surface, upon ivhich it is 
confined, is equal to the normal force which acts in this plane normal to 
the brachystochrone. 

At the beginning of the motion when the velocity is zero, 
there is no centrifugal force, so that the initial direction of the 
brachystochrone upon the surface coincides with thai of the tangential 
force. 

575. If the first and last points of the brachystochrone are 
so situated upon the given surface, that a line can be drawn 
through them, which coincides throughout with the direction 
of the tangential force to the surface, this line is the brachysto- 
chrone. 



— 334 — 

Hence, the brach/jstochrone upon the surface of revolution is the 
meridian line, ivhen both its extremities are upon the same meridian line, 
and the force is directed to a point upon the axis of revolution, or is parallel 
to this axis. 

576. In the general case of a surface of revolution and 
a force which is directed to a point upon the axis of revo- 
lution, let 

a denote the arc of the meridian curve measured from the pole, 
u the perpendicular from the surface upon the axis, 
o r the radius of curvature of the projection of the bracrrystochrone 
upon the tangent plane to the surface, 

and the proposition (333 20 ) is expressed by the equation 

-=DM tan 1 



Q 



T 



which gives 

r 

But the equations 



D s Sl _Z> s (v 2 ) 2 cot? 

Si — $2 v 2 Qt 



D 5 u = cos " = cos 1 cos Z, 

1 sin? cos Z 



D a - 



Vr 



give 



T~\ / ■ ,r\ U C0 

D s (u smf) = 



Qr ' 



and if A is an arbitrary constant, 

D s log v = D s \og (u sin z) 

Av= usm°=u 2 D s u x 
Av 2 =uv sin Z = u 2 D t " 

so that the area described by the projection of the radius vector upon 



— 335 — 

the plane of xy is proportional to the square of the velocity of the 
body. 

577. The equation (334 28 ) gives 

D„s = sec ' 



Sl (u-—A-ir) s/ [u 2 — 2 A 2 {SI — J2 )] ' 
tan?_ Av _A I 2{Sl—Sl ) 



j, „ tan? Av A I 2 

° x u ~u\J{u 1 — A 2 v' 2 ) u\ v? — 



2A 2 {tt — Sl ) 



578. If <3 is the angle which the radius rector makes with the 
axis, the preceding values give 



<t> ~ V « 2 — : 2A 2 {Si — si y 

D U __A / 2(Si-SZ,>)[f*+(Ptry] 
"** — u \ u 2 — 2A'(S2 — Sl ) ' 



When the forces are parallel these equations give 

u D.a 



D.s 



\j[_u 2 —2A 2 {Sl — Si. i) )'\ 



D « = ^_«. I 2(-^-^) 



2 A 2 {Si — Si u ) 



579. Upon the surface of revolution which is determined by 
the equation 

Bv = u 

in which B is an arbitrary constant, the value of % is by (33 4 28 ) 
constant, so that upon this surface the br achy stochr one makes a constant 
angle with the meridian curve. In the case in which 

A = B 

the brachystochrone becomes perpendicular to the meridian, and is 
a small circle, of which the plane is horizontal. 

Whatever is the value of B, the point at which v vanishes, 



— 336 — 

coincides with that at which u vanishes, so that at the pole of this 
surface the velocity vanishes. 

Upon any other surface of revolution about the same axis, the incli- 
nation of the brachystochrone to the meridian arc is the same with the 
corresponding inclination upon the surface of equation (335 22 ), at the com- 
mon circle of intersection of these tioo surfaces. Hence the limit of the 
brachystochrone upon a given surface of revolution is its circle of intersection 
with the surface of equation 

Av = u, 

and the brachystochrone extends over that portion of the given surface, 
which is exterior to the given surface, by ivhich the limits are thus defined. 

580. In the case of a heavy body, the surface of equation 
(335 22 ) is a paraboloid of revolution. When the velocity of a heavy body 
upon any paraboloid of revolution, of ivhich the axis is vertical and directed 
doivnivards, is fust sufficient to carry it to the vertex, the brachystochrone 
mahes a constant angle ivith the meridian curve; but ivhen the velocity is 
too small to carry the body to the vertex, the brachystochrone is a curve 
ivhich mahes an increasing angle with the meridian as it descends, and may 
sometimes become perpendicular to the meridian ; and when the velocity is 
more than sufficient to carry the body to the vertex of the paraboloid, the 
brachystochrone is an infinite curve, which is horizontal at its highest point, 
and diminishes its angle with the meridian as it descends. 

If the equation of the paraboloid is 

u 2 = 4 p z 

in which the axis of z is the downward vertical, the equation (334 28 ) 
becomes 

si »°= 4 v / 'fe( 1 -:?)]- 

If z is positive and 

p>iA* ff> 



the substitution of 







o o "7 






shr 


a 


•lp 


? 






q. 


= &o t« 


Lll 2 « 


■> 


Cos 


if: 


= + 2 - 


z-\-p 


+ <? 


P — 


q ' 



gives 

s = h sec a (p — q) [tp + Sin (jp) , 

in which the upper signs correspond to the case in which p is 
greater than q, and the lower to that in which p is less than q. 
In the case in which p is greater than q, the substitution of 



COS^ 111 = ; — -, 

z +p 

' 2 • P 1 

sin i - 



P-\-z» 

gives 



= — tan a i 

P- 



V / ( 1 -J- - ) 3^ u< — S; i/' — cot ^ y/ ( 1 — sin 2 / sin 2 if ) 

-^(-^,t)]. 

When ^ is smaller than q, the substitution of 

2 ~ z 

COS W S= : , 

T " + ? 
• '2 • <7 P 

gives 

I = tan a J (2+5) [§, y —^ 9?. ^ _|1- cot y y/ (1 — sin 2 / sin 2 1//) 

When 

43 



— 338 — 

the arc is 

s = sec a (z — z ), 

so that its inclination to the axis is constantly equal to a, and the brachys- 
tochrone is defined by the equations 

z = z sec 2 cp , 

"= tan a i /-(tan (p — </>). 
When 

the arc, measured from its cusp, is 

S = 3^ ^ 2 +^) f — (*« +P) 1 ] • 
and if 

the brachystochrone is defined by the equations 

p -\-z =p sec 2 / Cos 2 (p , 
^L5 = tan(|^ , + -2 5 »). 

tan j< \2sm2j' ' tan 2 r / 

when 

in which case the brachystochrone has a lower limit at which it is 
horizontal, the substitution of 

2 ^<f 

sec a ■= — — , 
2p 

-*■ cm" 1 r/ ' 



COS W = '-j- , 

T P + 9 



— 339 — 

gives, at the lowest point of the curve, where (p vanishes 

z = q, 
and for the value of s, measured from the lowest point, 
s = £ [p -f- q) cot a (sin (p -f- cp ) . 

The substitution of 

2 9 — z 

tan ijj = I — -, 



'0 



sin-" i 



'p+q. 

gives 



K-*^«v+*»<(-^.v)]- 



sin«y/[p (p-\-q)] L ' ' q~ ' ' q * \ y 

When O is negative, in which case the condition (336 30 ) is 
satisfied, the substitution of the equations (337 2 _ 5 ) with the lower 
sign gives the corresponding value of (337 7 ) for the arc measured 
from its upper limit, which corresponds to the vanishing of (p. 

When 

— 0o<2>, 
the substitution of 



i 



COS If 



r 



+ ? 



z-\-p* 



• 2 ' P -\~ ~0 

snr i = JL - 1 — ■ 

p — q 

gives 



I = tan a \J \- — - j I % { y — % i \\> -\- cos xj> y/ (cot 2 y -f- cos 2 i) 

1 /? — y \ j9 — ^ ' /J 

When 



— Zo>P 



340 — 



the substitution of 



cos 1/ 



; „, __ * + ? 



• 2 • P-\~ Z 

snr i — 



gives 

" = tan « w ( — - — -) \$i xp — 2F t - ijj -4- cos \\> \J (cos 2 i/; -|- cos 2 i) 

When 

the brachystochrone is defined by the equation 

•, = ton« [ v /(^ + ^|' k? ^g+^f]. 

581. In the case of the heavy body upon the paraboloid of revolu- 
tion in which the axis is vertical and directed vpivards, the br achy stochr one 
forms an increasing angle with the meridian as it descends and is perpen- 
dicular to the meridian at its lowest point. In this case, the inclination 
to the meridian is determined by the equation 

if (33625) is the equation of the paraboloid. By the substitution of 











sin 2 a = 


~ 2p ' 












q = 


= z Q tan 2 a 


■> 










Cos cp = 


_2z + />- 
P+<1 




9> 


vanishes 


at 


the 


lowest point 


where 





— 341 — 

and the value of the arc, measured from the lowest point, is 

s=% {}) -4- q) sec a ((p -\- Sin ip) . 
The substitution of 



tair w = 



snr i = -.— 



gives 



582. In the case of a heavy body upon a vertical right cone, if the 
vertex of the cone is assumed as the origin, and if 

a is the angle which the side of the cone makes with the axis, 

A 2 q cos a 

1\ = — ~2 3 

sin a 

Q = the angle which r makes with the axis upon the developed 
cone, 

the inclination to the meridian, the derivative of the arc and of & are 

. VC^'iO— r <>)] 



sin 



D r s 



r 
r 



v /[ r s_2r 1 (r — *•„)]' 

D 6 = - V[ 2y, i( r — r o)] 
When 

the substitution of 



r sj [r- 


— 2 r l (r — 


-*•«)] 


2 


ro>r 1 , 




sin 2 / 






an y 


r, cot i 




— r-V 




r 


= r sec 2 


£?, 



— 342 — 

gives 

cos 1 = cos i sec {x\j — i), 

s = r sin 2 i (cosec x}> — cosec 2i) — r x log (tan \ y cot i) , 
£ = — sin [-1] (sin i sin y) -\- sin i&i<p, 

in which the arc is measured from the cusps, at which point .- 

6=cp=1=0, xf>=2L 

This br achy stochr one extends to infinity from the cusp without ever becoming 
perpendicular to the side of the cone. The greatest angle which it makes 
with the side is i, and at this point of least inclination to the side 

tfj=i, r=2r , (p=z^n, 

6 = — i -j- sin * ^ ( h n ) . 
When 

2r =r 1 , 

the br achy stochr one is defined by the equation 

H = tan.-y(^-l)_Cot.-y(^-l), 



and the length of the arc, measured from the point of least inclina- 
tion to the side, is 

s == r — 2 r -f- r log (£■ — l) • 

When r is positive and 

2r <r 1? 
the substitution of 

Sec 2 /' = — , 

m I r i Tan P 

Tan w = + , 

T — r — r ^i 



gives 



— o4o — 



\ Tan ft Cosec \f> — 1\ log Tan h y> 



in which the arc is measured upon each branch from the point at 
which it is horizontal and the upper sign belongs to the lower 
branch and the reverse. The upper branch is finite, tvhile the lower 
branch is infinite, and the value of if) extends on the upper branch 
from 2 (1 to infinity, and on the lower branch from infinity to zero. 
For the upper branch the substitution of 



sin i = e 2 P, 



r — r = r sm i sir cp , 
gives 

<3 = 2 (1 -f sin i) [9?< y — <3V (sin i, y)] . 

Upon the lower branch the substitution of 



? 
r — r, 



sin i sin a \\> ' 
gives 

6 = 2 (1 -f- sin a) ^ (sin t, i/> ) . 

TF7i£« r vanishes, the equation of the brachjstochrone upon the de- 
veloped cone is 

r = 2 1\ sec 2 £ 6 , 

and the length of the arc is 

s = 2 r x tan £<3 sec £ <3 -|- 2 r a log tan (i tt -|- I 6) . 

When r is negative, the substitution of 

2 r — 4 sin i 



Cosec 2 p = 



r x (1 -[-sin i) 3 



' — r x Cos p 7 



— 344 



r x Cot /•> Cosec i/> — r x log Tan \ if 



in which the order of the signs and of the value of \\> is the same as 
in (343 4 ) with reference to the branches. The upper and finite branch 
of the brachystochrone lies in this case upon the upper and inverted portion 
of the cone. The formulae (343 n , 343 16 _ 19 ), apply to this case, in 
which it must, however, be noticed that the sin i is negative. 

583. When the solid of revolution upon which the heavy body moves, 
is the ellipsoid of which the equation is 



(i)+(i)=i> 



the inclination to the meridian is determined by the equation 

A uS l{Ai-z>) • 

The problem naturally divides itself into two cases. In the first case 
the velocity is more than sufficient to carry the body to the highest point of 
the ellipsoid, the brachystochrone is a continuous curve which is horizontal at 
its highest and lowest limits, and which, alioays running round the ellipsoid, 
is most inclined to the meridian curve at the point 

In the second case, the velocity is not sufficient to carry the body up to 
the highest point of the ellipsoid, and the brachystochrone is horizontal at 
its lowest point, but has cusps for its upper points. In each of these 
cases the length of the arc can be found by means of elliptic 
functions. If in the first case — z x and 2 are the coordinates of the 
upper and lower limits, or of the common intersections of the 
ellipsoid with the paraboloid of revolution of which the equation is 

u* = 2A*g (s-*o), 



— 345 — 

and if in the second case 2 2 refers to the intersection of the ellipsoid 
with the paraboloid, while — s 1 is the coordinate of the intersection 
of this paraboloid, inverted at the horizontal plane of u'x, with 
the hyperboloid of revolution, of which the equation is 

(i) -•(!.)=!> 

the derivative of the arc is 

"* b a x \ (*+*,)(«,-*)■ 

In the first case, when the ellipsoid is prolate, and 



** — Ai-AV 



*z -*-*• u 



the substitution of 



gives 



* v. . — ■?- 



When the ellipsoid is a sphere, of which the radius is R, the 
hyperbola (345 6 ) becomes equilateral, and the length of the arc, 
measured from the lowest point, is determined by the equation 

S 2 2-J-2, Z 

cos ■= = '-j- -'. 

In the first case (344 17 ), the substitution of 



V 




r •> 


cos/ 




sin \l> t 


sin 1^ 2 ' 



44 



— 346 
gives, for the sphere, 

u 2coa 2 ±\l) l jx /cos \p 2 -\- 30S tt> l s \ , 2 sin 2 \ ip, , 



sin \jj. 

,.2 1 



2^1 -yp /cos il> 2 -)- cos -i/>, s \ . 2 sin 2 \ xp^ ^ / cos i/> 2 '-j- cos ipi s \ 
i; 2 'I 1 — cos U> 2 '2 R/ ~T" sin i/) 2 * V 1 -f- cos i/; 2 ' 2 i^/ 



_4cos 2 f \p x (Ja /cos ii; 2 -|- cost/)! s \ cosi/^ — cos xp 2 ^ / s \ 
sin u> 2 ' \ 1 — cos iX) 2 ' 2 jff/ sin i/j 2 * \2 11/ 

tan [_1] costfr-l-cosift! 

\/ (sin 2 i/> 2 cosec 2 ^ -f~ sin 2 ti^ sec 2 ^) 

In the second case (344 23 ), the substitution of 

C0S 2a) = ^=^=^, 

Z 2 Z 



%2 Z n 

. o . Z 2 Z 

Sill 2 2 - 



gives, for the sphere, 

u Z,-\-E g* /COS Tpo cos V>0 \ z \ R (Jfi / C0S tyo C0S ^ \ 

x R sin i/; 2 ' V 1 — cos i/; 2 ' " / R sin u> 2 * V 1 -|- cos \p 2 ' ' / 

-1 + ^ POP / C03ip 2 — COS1//Q \ op ( R(l— COS !/■<,) VI 

_ i^os^L J '\ 1— cosi/, 2 '^/ i \« x + 22cos V , 3 >VJ 

cos ^-cosy, g. _ cogec ,- tan[ -l] /n j 1 ^.. ^ 

1 smi/i 2 ' v (1-p-cos-i tan- 9) 

In the case in which 

the brachystochrone is defined by the equation 

- = tan i w Tan [ - 1] sin -^ 4- tan [ - 1] J^A. . 

* ' 2i? ' tan|x// 2 

584. In the case of a heavy body upon any surface whatever, 
it follows from (329 3 ) that 

v 2 2g(z — z ) 

- = -^ = a cos I . 

Qt Qr J p r 



— 347 — 

If, then, N T is the normal to the brachystochrone drawn in the 
tangent plane, and extended to meet the horizontal plane from 
which the body must fall to acquire its velocity, the preceding 
equation gives 

N T ={z— z )sec 2 p =$Q T , 

or the tangential radius of curvature of the brachystochrone is twice the 
tangential normal ivhich extends to the horizontal plane of evanescent 
velocity. This proposition is given by Jellett. 

585. When the force is parallel to the axis and proportional to the 
distance from a plane ivhich is perpendicular to the axis, the surface of 
revolution of equation (335 22 ) is an ellipsoid when the force is attractive 
toivards the plane, and it is an hyperboloid of tivo sheets when the force 
is repulsive from the plane. 

586. When the force is directed toivards a fixed point and propor- 
tional to the distance from the point, the surface of equation (335 22 ) is an 
ellipsoid if the force is attractive, but if the force is repulsive, the surface 
may be an ellipsoid or it may be an hyperboloid of tivo sheets. 

587. When the force is directed towards a fixed point, and 
inversely proportional to the square of the distance from the point, 
the surface of revolution of equation (335 22 ) is defined by an equa- 
tion of the form 



K 2 =4 (I_i). 



588. Other conditions might be combined with that of the 
brachystochrone. Thus if the total length of the arc is given, the normal 
pressure to the brachystochrone is 

in which b is an arbitrary constant, and is dependent, for its value, 



— 348 — 

upon the given length of the arc. This constant is generally infinite, 
when the brachystochrone is a straight line. 

589. Under the condition of the preceding section, the equation 
of the brachystochrone, in the case of § 569, referred to the coordinates of 
(330 17 ) is 

In the case of § 570, this equation gives 



2aq = 



PiP 



(P2-W 

590. In the case of the parallel forces of \ 571, (347 28 ) gives 

\1 — oasmy 
When the force is constant, this equation gives 

a 2 sin * 
^ g{l — basinlf ' 

so that when 

ba>l, 

the curve has points of contrary flexure. 

591. In the case of § 576, and with the condition of § 589, the 
equation of the brachystochrone has the form 

, , , = u sin % = u 2 D ". 
l-\-bv " 3X 

The inclination of the curve to the meridian arc is therefore con- 
stant upon the surface of revolution, which is defined by the equation 

Bv = u(l+bv), 

and this surface has the same relation to other surfaces of revolution in 



— 349 — 

respect to the braclvjstochronc formed under the present conditions with 
those which are indicated for the surface of § 579. 

In the case of a heavy body, the equation of this defining 
surface of revolution is 

2g(,-B )==(^- r J. 

592. If the condition is a mechanical one, such that the total 
expenditure of action, defined as in § 308, shall be given, the normal pres- 
sure to the br achy stochr one is 

in which b is an arbitrary constant, and is dependent, for its value, 
upon the given expenditure of action. When this constant is in- 
finite, the normal pressure is equal and opposed to the centrifugal 
force. 

It is apparent, from the preceding equation, that under the 
action of finite forces, this brachystochrone cannot be a continuous 
curve, in one portion of which the direction of the normal pressure 
coincides with that of the centrifugal force, and is opposed to it in 
another portion. 

593. Under the condition of the preceding section, the equation 
of the brachystochrone, in the case of \ 569, referred to the coordinates of 
(330 17 ) is 



Si — £2 



l 2 \vJ 



In the case of § 570, this equation gives 

_ [l-f25a(r 2 -r 2 )] 2 y/^-r-g) 
*> ~~ 1 — 2ba(f — rl) * p lS /a ' 

594. In the case of the parallel forces of § 571, (3492, t ) gives 

■ Q ~^° — s i n 2 * 

[l + 2b(n — Si )J~ s ' 



— 350 — 
When the force is constant, this equation gives 

595. i« ^e c«se of § 576 «ho? «t$A the condition of § 592, the 
equation of the brachgstochrone has the form 



The inclination of the curve to the meridian arc is, therefore, 
constant upon the surface of revolution, which is denned by the 
equation 

Bv = u{l + bv 2 ), 

and this surface involves, for the present case, the properties of the defining 
surface of § 579. 

In the case of a heavy body, the equation of this defining 
surface of revolution is 

2B*g{z-z Q ) = i?\l + 2bg{z-z«)-]\ 

596. The brachjstochrone in a medium of constant resistance is 
entitled to special consideration. In this case, it is convenient to 
introduce the length of the arc as the independent variable. The 
equation of motion along the curve is 

v*=2S2 — 27cs, 

in which k is the constant of resistance. This equation must be 
combined with the equation 

(A*) 2 + (A*) 2 =1. 

If h ,"i and | fi are the respective multipliers of these equations in 



— 351 — 

the method of variations, the brackystockrone is defined by the differential 

equations 

1 
H'l = -s, 

_ A Si vD,v4- k 

D s a = —^ = — s -y±- - ; 

and by the following expression of the normal pressure directed in 
the opposite way to the centrifugal force 

D z 12 sin v -\- D x Q. cos v = - — . 

When k vanishes, the value of /a, is 

l 

and, therefore, the value of /x is the negative of the reciprocal of the ex- 
pression which is obtained for v token there is no resisting medium, and 
which is independent of the magnitude of the fixed force. 

597. When the force is directed toioards a fixed centre, the nota- 
tion of § 569 gives by (330 ]5 ) for the value of \i, 



PS/2 

598. When the forces are parallel, the equation (331 31 ) gives 
jtt in the form 

a= *-. 

COS V 

599. From the preceding equations, the equation of Ike brackys- 
tockrone of a keavy body in a medium of constant resistance has the form 



Q = 



JR sin v 



[1 — li cos (v J),,)] 



3 J 



in which R, k, and v are arbitrary constants. 



Qc: 



52 — 

600. In a medium of ivhich the laio of resistance is expressed 
as a given function of the velocity, the derivative equation of mo- 
tion is 

vD s v = D s n—V i 

in which V is a given function of v. The differential equations, by 
which the brachystochrone is defined, become, if i (i and ^ are the 
multipliers of (350 29 ) and (352 4 ), 

D s (ft sin v) = D x S2 D s ^ , 

— D s ({i sin v) = D z S2 D s fi t , 

— 1 - — vD s ii 1 + H , 1 D v V=0. 
The reduction of these equations gives 

I) sH ,=D s £2D sfh = D s (l-\-l h r), 

- P = - + Pi F; 

and the expression of the normal pressure to the brachystochrone 
becomes 



D z £2 sin v -J- D x & cos v = — ^ 



D s Si v s ii 



Fi Vv s -f- v 2 



QIX X V 2 J) V V Q' 



601. When the forces are 'parallel to the axis of z, the equations 
(352 9 ) and (352 17 ) give 



a 

sin v ' 



F>! 



sin v v 



— 353 — 

G02. These equations give for the br •achy 'stock -one of a hear?/ 
body in a resisting medium, 

j. i 7 y a 9 

J r j. i Ksin* Vv 7 

by which v is cleterniined in terms of v. The substitution of this 
value of v in the equation 

v D v v ir 

- — — = — g cos v — V , 

gives the equation of the brachystochrone in terms of o and v. The pre- 
ceding formulae include the results obtained by Jellett in his inves- 
tigation of this particular case. 

When V is inversely proportional to the velocity, the equation 
of the brachystochrone may assume the form 

2 h [h cos 2 — a) -f kf sin 2 (v — a) 

^ m-\-g cos v \_h cos 2 (v — a) -\- £] 

When V is proportional to the square of the velocity and has 
the form 

the equation of the brachystochrone is derived from the elimination 
of v between the equations 

, \ </ cos a q sin v 

cx»(y — a)=--^ ' W , 

! a (q cos v 7 \ To q cos a , ,1 /V/ cos v , 7 \ 



COS i 



& q sin i 

603. In these cases of the brachystochrone in a resisting 
medium, it is apparent that the condition (329 6 ) is usually violated, 
and that Euler, consequently, erred in extending this proposition to 
the case of the resisting medium. 

45 



— 354 — 

604. The determination of the form of the curve constitutes 
the principal feature of the general problem of the brachystochrone. 
But the nature of the curve may be given, and the problem is then 
reduced to one of maxima and minima, in which the various param- 
eters of the curve are to be determined. Euler has shown that 
there is a peculiar analytic difficulty in some problems of this class. 
A single example will illustrate this species of inquiry. 

Let the given curve be the circumference of a circle, of which 
the plane is vertical, and let the ball start from a state of rest at the 
upper point. If, then, 2 a is the angle which the line, joining the 
two points, makes with the horizontal line, and if 1% is the angle 
which the radius drawn to the upper point makes with the vertical, 
the equation for determining i is 

sec i \%>i ( i n) — 8; (2 a — i)~] — [cot 2 (i — a) -j- cos f\ 



[^(^)-^(2«-^)]+^ v / 



cos 2 a /sin2(i — «) 



sin 2 a 



0. 



THE IIOLOCHRONE. 



605. A curve, in which the time of descent along a given arc, 
is a given function of the arc, or of its defining elements may be 
called a holochrone. 

606. The problem of the holochrone becomes simple, when the 
forces are fixed, and the time of descent is proportional to a given power of 
the arc. Thus, if the time of descent is expressed by 

T s =As 11 , 

in which s is the length of the arc. Let 



B = ±V_pl[]r h{1 _ h 2-2n ) - i J 



— 355 — 

in which the upper sign corresponds to the case, in which n is less 
than unity, and the lower to that in which n exceeds unity. The 
force along the curve is 

S = — Bs 1 - 2 ". 
When 

n = l, 

the force along the curve is 

a it B 

l A 2 s s 

607. When the force is that of gravity, the equation of the 
holochrone of the preceding problem assumes the form 

^sintr = — B s l ~ 2n . 

608. If the time of descent admits of being developed according to 
integral ascending powers of s, the developed expressions of S and 
S2 S are obtained from the formulas 

&s = > 2 B s , 
8=D S Q S - 

in which the successive terms of P, are obtained from the equations 
represented by 

u r 
The second member of this equation is to be developed in 
form precisely as if V were the symbol of derivation, and in the 
result there must be substituted for P J=0 and F™jP s==0 , the values 

P, = o = cos</)P S=0 , 
r :P s=0 = (l — Bin«+*9) Z?P,_ . 



— 356 — 

609. When the forces are fixed, and the time of descent is a 
given function of the initial value of the potential, the problem of the 
holochrone can be solved by the method applied by Abel to the 
case of a heavy body. If A is the final value of the potential, in 
which the arbitrary constant is determined so that the potential 
may vanish with the velocity, the time of transit expressed as a 
function of A, assumes the form 



T — i- f B n* 



The integral, relatively to A of the product of this expression, 
multiplied by 



7ts/(Si— Ay 

is 

o a a 

1 f Ta = 1 f\ 1 f B ^ s 1 

n Ja \l (Si— A) n^-2j A i^ [Si — A] Ja y/ (A— Si)l 

But the notation 

l 

rh=f x (-\o g zy-\ 



with the familiar equation 



i 

x a-l 
u 

gives, by a ready reduction 



r x a ~ x rar(i—n) 

J x (i— X y 



f\ 1 f Si*- 1 ]_x a n _ ,_ nx a 

jAl(Si—A) l -"jQ (A — Si)"l~ a ^ % '~ asinnn' 

n n 



Note. — The notation (-SoG^) is substituted for that of (91 2 .i)> which was unwisely 
introduced instead of the usual form, which is here restored. 



— 357 



If the product of this equation multiplied by a cp («) is in- 
tegrated relatively to a, and if the function f x of x is defined by 
the equation 

f(<f,(cc)x*)=f x , 
so that 

*J a 



the integral gives 



sinjMr f [ 1 f 2>o/a_l _ r 



which, when 
gives by (356 16 ) 






a 

zu 



V(#-^)' 





The general relations between 5 and J2 complete the solution, 
and indicate the form of coordinates in which the solution should 
be finally exhibited. 

610. If the forces are parallel to the axis of z, 12 is a function 
of 0, and the elimination of s between (357i 5 ) an d the equation 

cos* z = D z s, 

gives this holochrone expressed in terms of the length and direction 
of the arc. 

611. If the forces are directed towards a fixed point, which is 
assumed to be the origin of coordinates, the elimination of r be- 
tween (357i 5 ) and 

cos" = D T s, 

gives this holochrone expressed in terms of the length of the arc 
and its inclination to the radius vector. 



— 358 — 
612. If T A , developed according to powers of A, is expressed 

it is evident that 



s=v /^[^^4 



613. An interesting case of this potential holochrone is obtained, 
ivhcn the body is supposed to approach the point of maximum potential 
along a given curve, and the required curve is to be such that the ivhole 
time of oscillation shall be a given function of the maximum potential. If 
s x denotes the given arc, the time of oscillation has the form 

rp _J_ f Dy (* + *.) . 

A yjtja \/{A — Si)' 
so that, by the process of § 609, 






*y 



In order that the two curves may be continuous, the direction 
of the given curve must coincide with that of the level surface at 
the point of maximum potential. But this direction may be given 
by an infinitesimal bend at the extremity of the curve, so that this 
is not a practical limitation of the problem. 

614. If the given time of oscillation is constant, the equation 
(358 18 ) assumes the form 

B(s-+- Si y=(2; 

and the compound curve becomes a peculiar species of tautochrone, which 
was investigated by Euler in the case of heavy bodies. 

615. When the forces are not wholly fixed but mag depend upon 



— 359 — 

the velocity, the problem of the holochrone becomes, to a certain extent, 
indeterminate. For, if 

W=0, 

is an assumed equation between s, t and v, such that / and s vanish 

together, but when v vanishes, the resulting equation between s and 

t assumes a given form corresponding to the given condition of the 

holochrone, the derivative of this equation gives, for the expression of the 

force along the curve, 

D t W+vD s W 
1 ~ D v W ' 

from which the time is to be eliminated bg means of the assumed equation. 

616. In most problems, in which the forces are dependent 
upon the velocity, the form of R is not unlimited, but is usually so re- 
stricted that 

R = R S + R V , 

in which R s is a function of s and represents the action of the fixed 
forces, while R v is a function of v and represents the resistances, to ivhich 
the bodg is subject. In this form of the problem, geometers have not 
made much progress towards its solution, although the case of the 
tautochrone, exhibited in this aspect, has been the occasion of much 
discussion and many difficult memoirs. 

617. If the equation (359 3 ) solved with reference to t, ac- 
quires the form 

i=T hm 

the expression for R is 

1 — rD.T..„ 



R 



V v T l)V 



which is essentially identical with Lagrange's most general formula 
in the case of the tautochrone. 



— 360 — 

618. If the equation (359 3 ), solved with reference to v, ac- 
quires the form 

v=V s , t , 

the expression for R is 

R = vD s l\ t -\-D t V ht , 

which formula comprises Laplace's general form of solving the taato- 
chrone. 

619. If the equation (359 3 ), solved with reference to &, ac- 
quires the form 

the expression for R is 

R-. 



v — D t S v , t 



620. When the equation (359 3 ) is presented in the form 

T-\-jS-{-V=0, 

in which T, iS, and V are respectively functions of t, s, and v, the 
value of R is 

D t T-\-vD s S 



R = — 



I)„ V 



But D t T is a function of t and, therefore, of S -\- V; it may, 
indeed, be any arbitrary function of S -j- V, so that if tf denotes 
this arbitrary function, R becomes 



R = — 



D v V 



621. When, in the preceding section, $ is changed into 
— log S and 



F=logy, 



— 301 — 
the value of R may be presented in the form 

R = vy(^) + v i I> s \ogJS; 

which is the same with a familiar formula of Lagrange for the case 
of the tautochrone. 

622. The cases, in which the formula (3G1 3 ) assumes the form 
(359 15 ) are easily investigated. For this purpose let 

V 

—~s> 

and the derivatives of (361 3 ) give 

D v R = D zl -f 2v D,\ogS= A «., 
DsD v R = — Z gDlx + 2zSD s D s \og!3=:Q', 

whence 

Dlz = c 2>S 2 D s D s \ogS=2a, 

in which a is any constant. Hence 

X = a s 2 -f- b z -f- e, 

in which b and e are constants introduced by integration. The 
value of R is, then, 

R — eS-\-bv-\-(a + D s S)^', 

so that, if h and H are constants, the final values of 8 and R are 

£ = «# + &«> + *»*; 

46 



— 362 — 

and this formula of Lagrange is restricted to the resisting medium, in 
which the resistance has the form 

a -\-bv-\-h v 2 

which was first remarked by Fontaine. 

The form of T, in this case, may be derived from the equations 

T 

c = z, 



I „„ I 7, I e 7, I n „T I „ „-T 



B t T=j = az + b + °-=b + ccT± 



ac 



■> 



which irive 



V (2 .a) cos [(, - 1) sj (2 c a - g)] = ?" + VffiA'^ 

_ 2eavS-\-beS 2 -\-bav i 
b v S-\- e S 2 -j- a v 3 

When v vanishes this equation becomes 

v/(26«)cos[(t — t)^{2ea — b 2 )~]=b, 

so that the interval t — t is independent of the length of the arc, 
and the curve is a tautochronc if % is also independent of s, which 
is the case when S vanishes with s, that is, when 

M=- a 7 . 
ii 

This condition is always observed, if the direction of the curve 
coincides with that of the level surface at its termination, so that in 
every case, this holochrone is essentially tcmtochronons. 
623. If, instead of (359 25 ) we suppose 

T= T 



and if i\> denotes an arbitrary function, the value of R has the form 



B==1 }p(T s , v )^-vD s T s , 



op o 



When 

T s>v = S V-\- S± 

in which S and iS\ are functions of s, and V is a function of v, the 
value of R becomes 

which includes Lagrange's formula. Forms of this kind may be 
indefinitely multiplied, without diminishing the difficulty of obtain- 
ing such as are new and not included in the investigations of § 622. 
624. A curious case of the holochrone is introduced, when the 
form of R is 

R = R s + R v + v 2 JS 1 , 

in which ^ is a function of s. The only case of (361 3 ), which can 
assume this form is easily proved to be that of (361 31 ) when JS 
is left undetermined. If, then, the factor of ^' 2 , diminished by a 
constant, is inversely proportional to the radius of curvature, the 
form of the resistance, by including in it part of the term c S, is 
that of (362 3 ) increased by a term proportional to the friction upon the 
curve. 

If the fixed force, in this case, is that of gravity, and the axis 
of z is vertical, and if v is the inclination of the radius of curvature 
to the axis of z, the first and last terms of R give, if k is the con- 
stant of friction, 

q g sin v -\- leg cos v 

e ' 

a-\-D s S=a— y - j^ = — (h-) r -)S, 

1 a e — h g sin v — h h g cos v 

l> (1 — P) g cos v 



— 364 — 

so that the curve determined by (36 1 29 ) is included in this form. This 
is a generalization of Bertrand's similar investigation with regard 
to the cycloid. 



THE TACHYTROPE. 

625. A curve in which the law of the velocity is given may 
be called a tachytrope. 

626. When the laiv of the velocity is given in an equation between 
the velocity, the space, and the time, the formidce of § 615 are directly 
applicable to the complete solution of the problem ; and all the subsequent 
transformations of these formidce may be applied to the present case. 

627. When the time is not involved in the equation (359 3 ), 
but the portion R v of the force R is given, the other portion R s is 
determined by the equation 

vD,W p 

from which v is to be eliminated by the given equation (3 5935). 
Euler has solved various cases of this tachytrope. 

628. One of the simple examples, solved by Euler, is when, 
in the case of a heavy body, 

R v = —kv m y 

and the velocity is to depend upon the arc in the same form as 
if the body descended in a vacuum upon an inclined straight line, 
so that the equation (359 3 ) acquires the form 

v 2 = hs, 
whence 

g sin v = R s =ih-\- Jc (h sf m . 



— 3G5 — 

When 

m=2, 

this equation becomes 

g sin v = \ h-\- khs, 

or the required tachytrope is a cycloid. 

629. Another simple and interesting example of this problem 
was proposed by Klingstierna and solved by Clairaut. It is that 
of a heavy body in a medium, of which the resistance is propor- 
tional to the square of the velocity, approaching the origin with a 
velocity equal to that which it would have acquired by falling 
in the same medium through a height equal to the distance of 
the body from the origin measured upon the curve. In this case 



9 /1 „— 2ks> 



R v =kv 2 

whence the equation of the tachytrope is 

D s z = 2c-* ks — 1, 



of which the integral is 



/c(s-\-s) = l 



c 



-2/fcs 



630. A simple example of the problem of § 627 is that in 
which the velocity is uniform. In this case 

JR S = — R v = a constant = D s 11 , 

so that in the case of a heavy body this tachytrope is a straight line ; 
in that of a constant force directed toivards a fixed point, it is a loga- 
rithmic spiral ; and in every case the sine of the angle, at ivhich it inter- 
sects each level surface, is inversely proportional to the fixed force ivhich 
acts at the point of intersection. 



— 366 — 

631. When the given forces are parallel to the axis of s, and 
the given equation (359 3 ) is expressed in terms of v and z, the 
equation of the tachytrope is 

(D z 12 sin v 4- R v ) D v W-\- D z Wv sin v = 0, 

from which v is eliminated by means of the given equation. Euler 
lias solved several cases of this tachytrope. 

632. If, in this case, the curve is to be such, that the velocity 
shall have a constant ratio to that which it would have acquired in 
a vacuum, the equation (366 5 ) assumes the form 

Z> £2 sin v = — - — — . 



If the resistance is proportional to the square of the velocity, 
so that R v has the form 

B v = — 7cv 2 = — 2ka(n-\r H), 

the equation of the tachytrope is 

sin vD z log (12-J- H ) = ^7,- 



633. When the given forces are directed towards the origin, and 
the given equation (359 3 ) is expressed in terms of v and r, the equation 
of the iachgtrope, in a medium of given resistance is 

(D r tt cos ; -f R v ) D v W-\- D r Wv cos r s = 

from which v is eliminated by means of the given equation. 

634. If, in this case, the curve is to be such that the velocity 
shall have a constant ratio to that which it would have acquired in 
a vacuum, the equation (366 24 ) assumes the form 

D r £2 cos r s = — 3-. 

s 1 — a 



— 367 — 

If the resistance has the form (36G 1G ), the equation of the 
tachytrope is 

cos: D r \og (11 ± II) = ^- a . 

635. When the law of the velocity, in a medium of known resist- 
ance, is given in a given direction, such for instance as that of the axis 
of x, and so given that 

2>COS* = W s>x , 

in which W sx is a given function of s and x, the equation of the 
tachytrope is derived from the equation 

(D s 12 + M v ) cos * — - sin s x = vD s W Si x -\- v cos % D x W s , x ; 

from which v is eliminated by the given equation. 

636. "When the velocity in the given direction is uniform, 
these equations become 

v cos^. = a, 

n? sin i 

Q — 



(A^+A)cos s r 

637. When the given force is that of gravity, and (i is the in- 
clination of the given line to the vertical, the equation of this tachytrope 
becomes 

or sin s 



(g cos (fi — y + £ v ) cos 3 £ ' 



This problem is solved by Euler in the case in which the given 
direction is horizontal and in that in which it is vertical. A special 
solution is obtained upon the hypothesis of a constant velocity ; in 
this case, the tachytrope is a straight line determined by the con- 
dition 

0cos(/J — *)+£„ = 0. 



— 368 — 

638. When there is no resisting medium, the equation (367 2 ,t) of 
the tachy trope becomes 

a 2 sin x 
*> g cos 3 s x Cos (fi— s x y 

When the line is horizontal 
and the equation becomes 

9 

a 
s gcos ]U J 

so that the tachy trope of this case is a parabola. 
When the line is vertical 

= 0, 

and the equation becomes 

a 2 sin * 

Q = r!> 

s - ^cos 4 i' 

so that the tachy trope of this case is the evolute of the parabola. 
With the notation 



b 2 = 



fl^sin/S 



2d 2 
the equation (368 3 ), expressed in rectangular coordinates is 

2 b \j (x -\-y cot /?) — b x = 2 cot § log [cot /3 -(- h y/ (> -f- y cot /?)] . 

639. If the resistance is proportional to the velocity, so that 

R v = — 7t y , 

and if the direction of the line in which the velocity is given is 
such that 

g COS/j :=ka, 

the equation of the tachytrope of a heavy body is 

a 2 ^ x 

x sin (i — y cos fi = — c~ir. 



309 — 



THE TACnYSTOTROrE. 



640. The curve on which the final velocity in a given resist- 
ing medium is a maximum, may be called a tachystotrope. 

641. In a medium in which the law of resistance is expressed as 
it is in § 600, the notation of that section gives for the differential 
equations of the tachystotrope 

D s O sin v) = D x 12 B s fa , 
— D s cos v) = D z 12 D s fa, 
v D s fa = fa D v V. 

The reduction of these equations gives 

D, fJt = D $ £2 D s fa = D s {fa V) , 

P = Pi V, 

and the expression of the normal pressure to the tachystotrope be- 
comes 

D z 12 sin v A-D x £2 cos v = /* = — =^ = l ^ n 



642. In the case in which the law of the resistance is ex- 
pressed by the formula 

V— k v m , 
the normal pressure becomes 

P m q 

so that the normal pressure has a constant ratio to the centrifugal force, 
which result was obtained by Euler in the case of a heavy 
body. 

47 



— 370 — 

643. Wlien the resistance is constant, the tachystotrope is a straight 
line. 

644. When the forces are parallel to the axis of z, the equa- 
tions (369 9 ) and (359 16 ) give 

a 

sin v ' 



a = Mi V= - 



645. The equation of the tachystotrope of a heavy body is ob- 
tained, therefore, by the elimination of v betiveen the equations 



V-. 



ga 



5 sin v — a cos j'' 



v D v v a a 

■ — = (7COSV y-. . 

Q " o sin v — a cos v 

646. When V has the form (36924), the equation of the tachy- 
stotrope of a heavy body is 



jt—. ro = h? (m a q sin vY 

(osmv — acosvy \ t/ \ / 



THE BARTTROPE AND THE TAUTOBARYD. 

647. The curve, in which the law of pressure is given, may 
be called a barytrope, and that barytrope, in which the pressure is 
everywhere the same, may be called a tautobaryd. 

648. When the pressure is a given function of the arc, which 
may be denoted by S, its equivalent expression, if F is the fixed 
force which acts in the direction /, is 

-—.Fean f = iS'. 

Q P 

and the differential equation of the barytrope is 

2R = D S [<> (#+ JFcosft] = 2F$m f p -\- 2 R v . 



Oil 

649. In the case of a heavy bod//, if the axis of z is vertical, the 
differential equation of this barytrope becomes 

{S-\-g cos v) D s q= — q D s S-\- og sin v -f 2 R v , 

from which v may be eliminated by means of the equation 

v 2 = (> ( S -\- g sin v) . 

In this case, the differential equation of the taidobaryd is 

(a -\-g cos v) D s q = Sg sin v -j- 2 7?^. 

650. When the resistance is constant, the equation of the barytrope 
of § 648 is 

o ( S + F co s / ) = 2 ( £1 -f //) -f 2 , JZ. . 

i;i #w c«sc o/" //^ /^#ry £o«^, this equation becomes 

SD v 8-\-gcozvD v 8 = 2ge + 2 11+ 2 s R v ; 
and that of the taidobaryd is 

[a-\-g cos v) b+3 Q == A \_g -{- a cos v -J- sin v \/(^ 2 — « 2 )]*, 
if J. is an arbitrary constant, 

and 



But if 

the equation of the taidobaryd is 

, \q, , sol 2 R v r _n'7 + f < !C0S1; 



— 372 — 

When there is no resistance, the tautobaryd of the heavy body is 
defined by the equation 

__ A 

^ ' {A-\-g cos*') 3 ' 

651. When &' vanishes, there is no pressure against the bary- 
trope, and this curve is that on which the body moves freely. 
Thus the equation of the barytrope of the heavy body becomes, 
under this condition, 

A 
^ ~ (cos vf ' 

ivhicli is that of a parabola. 

652. When the curve of the barytrope is given, the equations (370 27 ) 
and (370 31 ), determine the laiv of the fixed force ivhen that of the re- 
sistance is hioivn, or, reciprocally, that of the resistance, when the fixed 
force is known. 

653. When the forces are parallel to the axis of z, the equation 
(370 31 ) becomes^ 

AW.) + ^^ = 2JJ„ 

ivhich is applicable when the curve is given. 

When there is no resistance, this equation gives 

Fy cos 3 v = —J^[cos 2 vD s (#<>)] =— f v [cos 2 vZ> v (#(>)]. 

654. In the case of parcdlel forces, when the tautobaryd is a 
circle, and there is no resistance, the fixed force has the form 

F=-^ 

Q COS 3 V 

in which b and J 7 must vanish, if v can become a right angle. 






When the fixed force is that of gravity, and the tautobaryd 
is a circle, the expression of the resistance is 



Xv=— y(r— «)• 



655. In the case of parallel forces, when the tautobaryd is a 
cycloid of which the base makes an angle a with the direction of 
the parallel forces, and when there is no resistance, the equation of 
the cycloid being 

o = 2 R sin (v — a), 

the expression of the force is 

j-j a sin (r — a) -j- ^ a sin (3 v — a) -\- ^ a sin (v -\-u) -\-b 

2 sin (v — «) cos 3 v 

When b vanishes and a is a right angle, this expression is re- 
duced to 

F= i a cosecv, 

which coincides with Euler's solution of this example. 

THE STNCHRONE. 

656. The surface or curve which is the locus, at any instant, 
of all the bodies which start simultaneously from a given point 
with a given velocity, and move upon paths which are related by 
a given law, is called a sf/nchrone, and the given starting point may 
be called its dynamic pole. This class of loci was first discussed by 
John Bernoulli. 

657. If an integral of the motion of the body along one of 
the paths to the synchrone is obtained in the form 

W=0, 



— 374 — 

in which W is a function of the time, of the arc of the path, and of 
the parameters by which the relationship of the paths is expressed ; 
this equation is the required equation of the synchrone, if the time is as- 
sumed to be constant ; and it is referred to the system of coordinates, con- 
sisting of the described arc and the given parameters. 

658. If the only force is that of a resisting medium, and if 
the form of the path is given, and also the position of the dynamic 
pole upon it, but not its direction in space, the synchrone is obviously 
the surface of a sphere, of which the dynamic pole is the centre. 

659. If the body moves, without external force and without 
resistance, upon a straight line, which rotates uniformly about a 
given axis passing through the dynamic pole, the synchrone is a 
surface of revolution about the same axis, and it is defined by the polar 
equation (250 20 ) or (251 3 ) when p vanishes and t is constant. 

660. When the fixed forces are directed toivards a point, or ivhen 
they are parallel, the synchrone of bodies moving upon straight lines, is a 
surface of revolution, of which the axis is the line of action which passes 
through the dynamic pole. 

661. In the rectilinear motion of a heavy body, it is obvious from 
(255 13 ), that the polar equation of the synchrone has the form 

r = a cos^-j- b, 

which becomes a sphere, when b vanishes, that is, when the initial velocity 
vanishes. 

662. In the rectilinear motion of a heavy body through a medium, 
of which the resistance is proportional to the square of the velocity, the 
polar equation of the synchrone has the form, 

Ac r =Cos(Bcos ir z ). 



375 — 



TriE SYNTACHYD. 



663. The surface or curve which is the locus of all the points, 
at which bodies have the same velocity, when they move from a 
given point, with a given velocity, upon paths which are related by 
a given law, may be called a syntachyd. 

664. If an integral of the motion of a body along one of the 
paths which proceed to the syntachyd is obtained in the form 

W= 0, 

in which W is a function of the velocity, of the described arc, and 
of the parameters, this equation is that of the syntachyd in the 
same form of coordinates with those in which the synchrone of 
§ 657 is expressed. 

665. In the case of §658, the syntachyd coincides with the syn- 
chrone. 

666. In the cases of §§659 and 660, the syntachyd is a surface of 
revolution about the same axis ivith the synchrone. 

667. When the action is exclusively that of fixed forces, the syn- 
tachyd is a level surface. 

668. When a heavy body moves upon a straight line, on which there 
is a constant friction, and through a medium of ivhich the resistance is 
proportional to the square of the velocity, the equation of the syntachyd is 

c- 2hr — A = B cos (; + «), 

in which the notation of § 515 is adopted, A and B are constants 
and 

a = g tan a . 

669. When a heavy body moves upon a straight line, on which 
the friction is constant and through a medium of ivhich the resistance is 



— 376 — 
proportional to the velocity, the equation of the syntachyd has the form 

670. When the body moves upon a line on which the friction is 
constant and through a medium of which the resistance is proportional to the 
square of the velocity, the equation of the syntachyd, expressed in the form 
of coordinates of § 657, is 

(kv 2 -\- A) c 2ks =f s (D s Q c 2sh ), 
which coincides with Jacobi's investigation of this case of motion. 



A POINT MOVING UPON A FIXED SURFACE. 

671. Among the various forms, in which the motion of a point 
upon a fixed surface, with fixed forces, can be discussed, that of the 
principle of least action is here selected. In this case, therefore, 
the whole amount of action, denoted by 

v =f,"' 

is to be a minimum. If, then, the equation of the surface is 

■L=0, 

if rectangular coordinates are adopted, if ^ is the multiplier of the 
preceding equation of the surface, and ^ that of the conditional 
equation 

the equation of the path of the body, with reference to either axis, is 
D x v-^-\i x D x L — D s (fiaf)=0. 



— 377 — 

The sura of these three equations, multiplied respectively by x' , y', 
and z\ is 



D s v = D s \i, 



or 



Whence 



D x v -f ih D X L — D S (v x) = . 



672. If the tangent plane to the given surface is assumed, at 
each instant, to be that of xy, and if the axis of y is taken normal 
to the path of the body, the preceding equation becomes, if ()j de- 
notes the radius of curvature of the projection of the path upon the 
tangent plane, 

so that the centrifugal force of the body in the direction of the surface to 
which it is restricted is equal to the normal pressure upon the path in the 
direction of the tangent plane. 

673. When the direction of the force is normal to the surface, which 
is the case ivith the level surface, or when there is no force, the path of the 
body is the shortest line which can be drawn upon the surface, and coincides 
ivith the br achy stochr one. 

674. When the velocity is constant, the equation (377 13 ) expresses the 
condition that the body may move upon the intersection of a level surface 
ivith the given surface. In this case q x is the radius of curvature of this 
intersection, and D y S2 is the whole force in the direction of the 
tangent plane to the surface. 

675. When the velocity is a given function of the parameter of 
the level surface, the equation (377 13 ), with the notation of the pre- 
ceding section, expresses the equation of a surface over which the 
body moves upon the intersection of this surface with the level 

surface. 

48 



OHrQ 

O/b 

676. When the force is directed towards the origin, and the given 
surface is a plane passing through the axis, the equation (377i 3 ), com- 
bined with (316 18 ), gives in the notation of § 569 

D,..Q _ _ 2 _ 2 D,.p 

Si — Si ~ q sin r s ~ p ' 

of which the integral is 

n Q !Zi 1 ? ,2 I T) «j2 

'-"-• "i-o y 2 " 2 -t-'t » • 

Whence, if (p is the angle which r makes with the axis, 

1 2 T\ 9 

ir*D t y=pl. 

But 2 r 2 c/g) is the elementary area described by the radius vector in 
the instant dt, and it, therefore, follows that the area described bg the 
radius vector is proportional to the time. 

The equation (378 n ), combined with that of living forces, gives 

D t s*=py + r i I) t <p* = I> t r i +^=2(£2 — S2 Q ), 

r ^ — D t r~ rs /[(2r 2 (J2 — i2 )_ ±p*]> 

Whence 



2 pi 

^ " " I- r v/ [2 i*(J2 — ^2 ) — 4$] : 



-I 



which is the polar equation of the path of the bodg. That this equation 
can be obtained by integration by quadratures, is a simple case of 
the principle of the last multiplier. 

677. When the potential of the force has the form 



Q = — 



and the initial velocity is such that 



o — o 



— 379 — 

or 

„2 9 f) — \l 

the polar equation of the path of the bod// is 

plr n -' = ^(Ut)sm [0 — l)(fjp — «)], 

which was given by Eiccati. 

If 

n= i, 

the law of the force is that of gravitation, and the path is a parabola of 
which the origin is the focus. 

If 

n = 1, 

the attractive force is inversely proportional to the cube of the radius vector, 
and the path is a logarithmic spiral, which was proved by Newton. 

If 

n = |, 

the attractive force is inversely proportional to the fourth power of the radius 
vector, and the path is the epicycloid formed by the exterior rotation of a 
circle upon an equal circle, which was proved by Stader. 

If 

n = 2, 

the attractive force is inversely proportional to the fifth poiver of the radius 
vector, and the path is the circumference of a circle, which was proved by 
Newton. 
If 

the attractive force is inversely proportional to the sixth poiver of the radius 
vector, and the path may be called the trifolia of Stader, by whom it was 
investigated. 



— 380 — 

If 

n = d, 

the attractive force is inversely proportional to the seventh poivcr of the 
radius vector, and the path is the lemniscate of James Bernoulli, which 
was proved by Staler. 

If 

n == — 1 , - 

the repulsive force is proportional to the radius vector, and the path is 

an equilateral hyperbola. 

When 

n<l, 

r becomes infinite when (<p — a) vanishes, which was remarked by 
Staler. 

678. When the values of £2, S2 and p x are such that, if R is 
an integral function of an integral root of r, 

, R = sJ[2r*(tt-S2 )-±pf], 

the expression of (p in (378 22 ) admits of integration. For if the 
integral root of r is denoted by 

mi 

9\= \Jr, 

and if the notation of the residual calculus is adopted, the equa- 
tion (378 22 ) becomes 

log(Vr— r } ) 



o 2 C 1 o 2 r lo g(V r_ 

V = 2 m &Jr ^R = 2 "A ^ -%H 



679. An example of the preceding section occurs, when m 
is unity and 

R=.af" -\-br -\-c, 



— 381 — 
which corresponds to 

1 ' r ' 2 r 2 ' 

S2 = — -H 2 —ae, 
and an attractive force of the form 

2 i i l> e i e 2 -\-4n\ 

— a 2 r — «* + 75-H — -75-^ • 
In this case the value of 9 is 

cp — a = - log -75 . ., — 777 tan 1 iJ -7-7-7 — t?, , 

' e ^ H e\J(±ae — b 1 ) \J(4ae — b 2 )' 

Pi -1 r 2 1 2 5pf ^ r _ 11 2ar-\-e 

= — loff-^-1 , .,., , — r Tan 1 ] -7-777, — l . — r, 

e ° H ' e\J (b 1 — 4 a e) \J (V — 4 a e) 

When b vanishes, these expressions become 

J2 = — ae, 

the attractive force is 

2 1 e 2 + 4/^ 

and the equation of the path is 

When e vanishes, the expressions become 

S2 = ia 2 r 2 + al>r-\- 2 -4, 



n» = — u\ 

the repulsive force is 



2 1 7 4: Pi 

a r-+-ab V » 

1 r > 



— 382 — 
and the equation of the path is 

« 5 (<P — a) = locr («4--) — • 

2 ap{ vr ' & \ ' rf r 

When Z» 2 — 4 a e vanishes, the equation of the path is 

G80. Another example of §678 occurs when 

B = ar.+ b+ e p 
which corresponds to 

n =—u 2 , 

and an attractive force of the form 

a b J*_|_2ae + 4;»J , She , 2 e 2 

The equation of the path is ■ 

2ar-f & = v^(4ag — y)tan[ ^ (4 ^7 y) (y— a)] 

When a vanishes, the value of I2 vanishes, the attractive force is 

tf + Apt , 3be , 2e 2 

and the equation of the path is 

log (br-\-e) = j-i(<p — a). 
When P — 4 a e vanishes, the equation of the path is 

2ar + b= 4p ' 



a — (jp 



— 383 — 

681. Another example of § 678 occurs when 

fi = Ar'l + B, 
in which case 

and the equation of the path is 

^|(a-9)) = log(l-ffr-^). 

682. The forms, in which (378 22 ) admits of explicit integration 
without any special determination of S2 and p, are included in the gen- 
eral expression 

in which h is two, or the negative of unity, so that 12 only consists of tivo 
terms, of which one is 

b_ 

and the general form of the centred force consists, therefore, of two terms 
of which one is inversely proportional to the cube of the radius vector, and 
the other may be either directly proportional to the radius vector, or in- 
versely proportional to the square of the radius vector. 

683. In general, it is apparent that the addition of a term to 
the central force, which is inversely proportional to the cube of the 
radius vector, does not augment the difficulty of determining the 
path of the body. In any equation of a path of a body described 
under the action of central forces, which is expressed by the elements 
(p — a,v and t, and which may also involve the constant p 1? the multi- 
plication of the angle w — a , and of p^ by the factor 



*=^-^) 



— 384 — 

gives the equation of the path, when the central force is increased hy the 
term 

b_ 

684. When there is no force the path is a straight line, so that 
ivhen the central force is inversely proportional to the cube of the radius 
vector, the polar equation of the path is 



r cos [B (cp — «)] = B p\ i/ ■ 



If the force is repulsive, B exceeds unity, the path is convex to the 
origin, and its convexity increases with the increase of the repulsive 
force until it terminates in a straight line. If the force is attractive, 
and B 2 positive, it is less than unity, the path is concave to the 
origin but of infinite extent, and the concavity increases with the 
increase of the attractive force until it terminates in the reciprocal 
spiral of Akchimedes. If the force is attractive, B 2 negative and 
S2 positive, the equation of the path is 



r Cos \B (9 — a) \j — 1] = B p\ J ^ 



iV 



so that the greatest distance of the path from the origin is limited, 
and the path is a spiral about the origin in which it terminates, at 
each extremity, through infinitely compressed coils. If the force is 
attractive, and B 2 and 12 negative, the equation of the path is 

rSin[B(cp-a) s /-l-]=Bpl s J±, 

so that the curve extends to an infinite distance from the origin at 
one extremity, and terminates in an infinitely condensed coil about 
the origin at the other extremity. In these three cases, the formula 



— 385 — 
for the time which corresponds to (384 9 ) is 

t = — £*tan[i(g> — a)], 

the formula for (384 20 ) is 

t ===■ Bp ^~ 1 Tan [i? (y - a) V - 1] , 
and that for (384 27 ) is 

^ = ^=- 1 Cot[5( 9 )-a)v/--l]. 

This law of central force has been discussed by several geometers, 
and, with peculiar regard to the special cases of the problem, by 
Stader, whose results coincide substantially with those of this 
section. 

685. When the central force is proportional to the radius vector, 
the path is a conic section of ivhich the centre is at the origin. It is an 
ellipse, if the force is attractive, and an hyperbola, if the force is repulsive. 
In the case of the ellipse, if a point were to start from the ex- 
tremity of the major axis at the same instant with the body, and 
move upon the circumference of which this axis is the diameter, 
with such an uniform velocity as to complete its circuit synchro- 
nously with the body, the body and the point are always upon a 
straight line which is perpendicular to the major axis. For dif- 
ferent ellipses, the time of description is proportional to the square 
root of the area. In the case of the hyperbola, if a catenary is 
drawn through the extremity of the transverse axis, in such a 
position that this axis is the direction of gravity, while its ex- 
tremity is the lowest point of the catenary, and of such a mag- 
nitude that the radius of curvature of the catenary at this point 
is equal to the semi-transverse axis, and if a body starts upon the 

49 



— 386 — 

catenary simultaneously with the given body, and proceeds in such 
a way as to recede uniformly from the transverse axis with a 
velocity equal to that of the given body at its nearest approach 
to the origin, the line which joins the two bodies will always re- 
main perpendicular to the transverse axis of the hyperbola. 

686. When in addition to the term, which is proportional to the 
radius vector, the central force has a term inversely proportional to the 
cube of the radius vector, the path can be derived from the preceding 
section by the principle of \ 683. 

When the term which is proportional to the radius vector is 
attractive and expressed by 

a r, 
the polar equation of the curve is 

i^- 1 -f I2 = v/ \fl\ —ia B 2 pf] cos [2 B(cp — a )] 

= v / [S2l — 4:aB 2 pf] Cos[2B((p — a) y/— 1] 
= y/ [4:aB 2 p\ — 12 2 ] Sin [2B((p — a) y/— 1]. 

When a is positive, therefore, the path does not extend to infinity, 
although when B 2 is negative it is compressed at each extremity 
into an infinite coil. But when a is negative, the term propor- 
tional to the radius vector is repulsive, and the curve extends to 
infinity if B 2 is positive ; but if B 2 is negative the curve is limited 
if S2 is negative, or it may necessarily extend to infinity if S2 is 
positive. 

In the special case of 

tan (2 n Bn) p\ 



2 B ii 



y/ — a. 



the curve is asymptotic to itself. 

687. When the central force is inversely proportional to the square 



— 387 — 

of the radius vector ivliich is the law of gravitation, the path is a conic 
section, of ivhieh the origin is the focus. When the force is attractive, the 
path is an ellipse if S2 is positive, a parabola if S2 vanishes, and it is 
that branch of the hyperbola ivhieh contains the focus, if S2 is negative. 
But ivhen the force is repulsive, the path is that branch of the hyperbola 
ivhieh does not contain the focus. The farther consideration of this law 
of force is reserved, in this connection, for the Celestial mechanics. 

688. When in addition to the term, ivhieh is inversely proportional 
to the square of the radius vector, the central force has a term inversely 
proportional to the cube of the radius vector, the path can be derived 
from the preceding section by the principle of § 683. 

If the term of central force, which is inversely proportional 
to the square of the radius vector is 



the polar equation of the path is 

i^_ a = v /( a 2_8/2 i? 2 ^)cos[i?(9) — «)] 

= y/ ( a 2 _ 8 i2 B 2 p\) Cos [B (cp — a) y/ — 1] 
= yj ( 8 £2 B 2 p\ — a 2 ) Sin [B (9 — a ) y/ — 1] , 

when S2 is positive, therefore, the curve is finite ; it returns into 
itself if B 2 is positive, but if B 2 is negative it terminates at each 
extremity in an infinitely compressed coil about the origin. When 
I2 is negative, one portion at least of the path extends to an in- 
finite distance from the origin ; if, moreover, a is positive and B 2 
negative, but such that 

a 2 >8S2,B 2 pi, 
another portion of the path is finite and terminates in the origin, 



— 388 — 

through an infinitely compressed coil, while the two infinite por- 
tions commence in such a coil ; if the negative B 2 is such that 



d i <8S2 B 2 p\ 



or if a is negative as well as S2 , the curve only consists of the 
portion which extends from the coil to infinity. The time may be 
computed by the three formulas, which correspond to the three 
forms of (387 M ), 

Tan [^ (^ 2 o + \/ y&pl + ii2 «) V Sln V* (y-°)V-l])] 

= -]££»%* Tan[^( y -. )V /-1] ? • 

rnt^^^o + v/^-^^rCos^^-.)^-!])] 

_ a — y/ (8 .<2 #>* — <) p fl p, . , -,-, 

--7(Z87VB^) b0t|_ 2 ^(9— aJV — 1J, 

the upper of the double forms of the first member applies to the 
case in which S2 is positive, and the lower to that in which /2 is 
negative. This case was partially developed by Clairaut. 

689. The principle of § 683 may be extended to § 677, and 
among the resulting curves, that in which n is 2, deserves to be 
noticed from its simplicity, the equation of this case is 

f 1 r = ^smlB(cp-a^=^ 1 SmlB( ( p--a) s l-l-]. 

690. The laiv of central force, for which the integrals, involved in 
the equations of motion, can be expressed ly the elliptic forms without 



— 389 — 

any special determination of S2 and p 1? may he reduced to two general 
forms of algebraic polynomial besides other fractional forms. 
The first of these forms is 



— 3 



F=b i r im - 3 -\-b 3 r 3m - 3 + b 2 r 2m - 3 -\-b 1 r m - 3 -{-br 

in which m is either 2, 1, -f, or -|-. 
The second form is 

F= b, r 2 '"- 3 -|- b 3 r m ~ 3 -f h r~ z + h r~ m ~ 3 + b r ~ 2m ~ 3 , 

in which m is either 1 or 2. In each of these cases the term which 
is inversely proportional to r must be omitted. 

691. In the first case of the preceding section, when m is unity, 
the equation (07822) acquires the form 

2 pi 



'" J-^V / ( a ^ 4 + «3 



z 3 r 3 -\- a 2 r 3 -f- a x r -\- a) ' 

It is obvious from inspection that whenever 

is positive, a portion of the curve extends to infinity; but when- 
ever « 4 is negative, the curve is of finite extent. It is also apparent 
that whenever 

a = — 4:B 2 pt, 

is positive, a portion of the curve terminates in an infinitely com- 
pressed coil about the origin, that no portion of the curve can ap- 
proach the origin except through such a coil, and that when a is 
negative, the curve does not pass through the origin. 
If all the roots of the equation 

a 4 r 4 -\- a 3 r 3 -f- a 2 r 2 -f- a x r -J- a = , 

are imaginary, « 4 and a must be positive, and the curve extends 



— 390 — 

continuously from the origin to infinity. If the moduli of the roots 
are h and h ly and the arguments a and a 1} and if the following 
notation is adopted 

I _ 2 « 4 (A» + MY — 4a 
P "I" ? — 2 fll — a 3 (A 2 + A?) ' 

_ 2 aa s — (h 2 -\- Itf) a x a i 

A 2 =(p — lif + 4 jk> h sin 2 £ a , 
A\ = (j» — y^) 2 + 4 jo /^ sin 2 £ a x , 
i? 2 = (y _ hf -\-iqli sin 2 £ a, 
i? 2 =(^ — A 1 ) a +4 ? A 1 sin a ia 1 ,- 



cos^: 



'^i (? — *■)' 

A X B 

a~b x > 



and if $ is the value of when r vanishes, the equation of the 
curve is 

AAJM(i-fi**b ) ( _^ = V(l-si^-sin^ ) og2 + . n2 j 

2p\{p — £) 2 cos 2 o xr ' (p — q)coa-d yi u ' u u/ 

-j^y ( X - sin2 /sin2 <W ^ (- cosec 2 d , 6) 

, y/ (1 — sin 2 « sin 2 (9) — y/ (1 — sin 2 i sin 2 O ) _ 
°~ y/(sin 2 — sin 2 o ) "' 

and the expression of the time is 

A A x q 2 tan d (t — r) / p 2 + <? 3 cos 2 i tan 2 # y (p cot 2 # + #) y/ (/> 2 + <? 3 cos 2 i tan 2 # ) 

~¥Xl—p){l — r ) V p 2 cot 2 o + ? 2 (? + '•)(/ cot 2 ^ + ? 2 )f 

<7 G° 2 + .P ?) V (P 2 + ? 2 cos2 * tan2 ^o) 



+ 



(q — r) tan 3 6 {f cot 2 6 + ? 2 )> 



I i sl{.(P 2 c °t 2 ^o + ? 2 ) (1 — sin 2 t'sin 2 0)] — y/ ( jo 2 cot 2 # + f? 3 cos 2 i) 
' °^ " ~~ VO 2 cot 2 O sin 2 — q 2 cos 2 l?)~ 



— 391 — 
The elliptic integrals disappear when 

which case has already been discussed in § 686. They also disap- 
pear when the imaginary roots are equal, in which case 

a 2 a 2 

- = - 1 = 4 « 2 — 8 \I (a a 4 ) : 
a 4 a tv*/' 

so that if 

fi=2a 3 « 4 r 2 -f- a| r -J- 2 «! « 4 , 

the expressions for cp and t are 

<p__ a= ZLW^ 2o »>? tari- 1 ! (4 g j - r +^VJ!l_ 

* \J a ° R y/[a 1 a 3 a 4 (16 a a 4 — ai0 3 )] y' [a 3 (16 a or 4 — cr x a 3 )] ' 

^ — t = — loa- 72 _ t / f g ' q » ) tant-^ ( 4a *r + « 3 )vS _ 

2y/« 4 ° V \a 4 (16aa 4 — a x a^)/ y/[a 3 (16aa 4 — ^03)] 

When two of the roots of the equation (389 29 ) are real and 
two are imaginary, if both the real roots are negative, « 4 and a 
must be positive, and the curve extends continuously from the 
origin to infinity. If one of the real roots, denoted by r lf is posi- 
tive, and the other, denoted by r 2 , is negative, and if a 4 is positive, 
the curve extends to infinity at each extremity, and i\ is its least 
distance from the origin ; but if a 4 is negative, the curve is finite, 
terminates at each extremity in the origin, and i\ is its greatest 
distance from the origin. If both the real roots are positive and 
if a 4 is also positive, the curve consists of two portions, one of 
which extends to infinity at each extremity, and the greater real 
root r 1 is its least distance from the origin, while the other portion 
is finite, terminates at each extremity in the origin, and r 2 is its 
greatest distance from the origin ; but if « 4 is negative the curve 
consists of a continuous portion of which r x is the greatest, and r 2 
the least distance from the origin. If h is the modulus and a the 



— 392 — 

argument of one of the imaginary roots, the following notation 
may be adopted. 

7\ — h c a v_1 = A tan !e d 3 v ~ 1 , 
r x — hc~ a * / - 1 = Atari h c-P^- 1 , 

r 2 — hc a " / ~ 1 = A cot ie (fii" / ~ 1 } 
r 2 — hc- a - / - 1 = Acot h c~ 8 i v_1 , 
TiCot =|e = /cot £y, 
r 2 tan %e = l tan £y . 

When a 4 is positive, if $ and i are determined by the equations 

i=i„Jll(P — ft) 

tan^cotie = y/(^^), 
or 

r sin y cos y — cos 9 sin i (8 -\- y) sin ^ (d — y) 

I sin e cos ?. — cos 6 sin \ (0 -\- e) sin ^ (0 — s) ' 

the equation of the curve is 

IA sin£(m— «W«4 1— cos£Cosy~ . . \ rf» / • 2 • a • a\ 
~~ — — = — — '-w^— cot y (cosy — cose)^ — sir/sin 2 ^) 



cos y — cos £ 



', ^ r_ 1] sin vsin^y'[l — sin 2 isin 2 y) (1 — sin 2 z'sin 2 #)] 
y/(l — sin 2 1 sin 2 j') 1 -)- cos y cos d ~\- sin 2 i sin 2 y sin 2 

and the value of (t — t) is derived from that of (cp — a) by multi- 
plying by g3 and interchanging y and s. It is apparent that e is 
obtuse and exceeds y, and that upon the finite portion of the 
curve 6 extends from zero to y, while upon the infinite portion, 
it extends from £ to n. 

When « 4 is negative, if & and i are determined by the equations 

tan i<3 cot 2« = v/ (7-377 > 



— o'Jo — 

or 

r sin y 1 — cos y cos d 

I sin s 1 — cos s cos d ' 

the equation of the curve is 

I A sin £ sin 2 y (<p — aW — a 4 • <> err a \ / \ jw / , i\ 

j-i — = sin 2 / 9^ 6 -J- (cos y — cos e) ^ (cot y, d) 

I (cos j' — cos s) cos j' , |-_j] / /cot 2 2' -\- sin 2 A 
~T ^(cot 2 y + sin 2 t) y \cot 2 -j- cos a t7 > 

and the value of (t — t) is derived from that of (cp — a) by multi- 
plying by g"— 2 ^ n d interchanging / and e. 

The elliptic integrals disappear when the two real roots are 
equal. In this case, a 4 is positive, and the curve is continuous from 
the origin to infinity. With the notation 

R 2 = r 2 -\-h 2 — 2 r h cos a = (r — h cos a) 2 -J- h 2 sin 2 a , 
R\ = r\ -\-h 2 — 2 )\ h cos a = (r 1 — h cos a ) 2 -\- h 2 sin 2 a , 

the equation of the curve is 

ri(q> — «) y/a 4 1 m „ n [-i] &— rcos« _ J_ m [_i] h 2 — h cos a (r-f-ri) -f r ^ 

2^ 2 -A- 1 "' 111 i? i?i " iJi?! ' 

and the expression of the time is given by the equation 

(*_ T)v /« 4 = S in[-«^=^-£ Tan^ ^ 2 -^ cos " ( ;+ ri) + ^ 1 . 
v - ' T A sin a K x li M 1 

When « 4 vanishes, if r x is the real root of the equation (389 29 ), 
the curve consists of a single portion which extends from the 
origin to infinity when i\ is negative, in which case a z is positive. 
But if 1\ and a 3 are both positive, the portion extends to infinity, 
and r x is its least distance from the origin ; if r± is positive while 
a 3 is negative, each extremity of the curve terminates in the origin, 
and r x is its greatest distance from the origin. 

50 



— 394 — 
When « 3 is positive, if 6 and i are determined by the equations 



tan 2 1 6 



a Atan±e~ B* ' 



the equation of the curve is 

, ™ B(q> — a)y/a a _ ota n + B' gp [ fa — B-f .1 

j B(r x -Bf [_!] sing S /(rf+^-2^r 1 cos2Q 

~W[ r i( r i+ jB4 — 2^^(50821)] 2^ V / L r i( 1 — sin 2 zsin 2 0)] ' 

and the expression for the time is 

(*_ T ) v /0 8= (j_|_ J B)g?.d_2i?g^ + 2i?tan^ v / ( 1 — smS'sin 2 ^. 

When « 3 is negative, if 6 and i are determined by the equations 

i=h{n— ft), 
tan 2 ^ = ^-, 
the equation of the curve is 

■ ^(^,+-g 2 ) 2 Tflti^ 1 ; — * ^ (rf ± Bi + 2B2r i cos 2 

' \/[r 1 (ri-\-B i -\-2B 2 r- 1 cos2i)'] 2B^ [r x (l— sin^'sin 2 0)] ' 

and the expression for the time is 

(zf — t)V — ; « 3 =(^— ^)^^H-2^M — 2^tan^v / ( 1 — sin 2 asin 2 d). 

When all the roots of the equation (389 29 ) are real, if, beginning 
with the greatest, they are arranged in the order of algebraic 
magnitude, they may be denoted by r ly r 2 , r z , and r 4 . If they are 
all negative, the curve consists of a single portion which extends 
from the origin to infinity. But if r x is the only positive root, the 



— 395 — 

curve consists of a single branch, which extends by the same law 
as that expressed in (391 20 ). If r } and r 2 are positive, while the 
other two roots are negative, the curve consists of one or two por- 
tions, according to the same principles which distinguish the forms 
of (391 25 ). If r 4 is the only negative root, and if # 4 is positive, 
the curve consists of two portions, one of which extends to in- 
finity, and ri is its least distance from the origin, while the other 
portion is finite and limited by the circumferences described about 
the origin as centre, with r 2 and r s as radii ; but if a 4 is negative, 
one portion terminates, at each extremity, in the origin, and r z is 
its greatest radius vector, while the other portion is contained be- 
tween the limiting circumferences of which r x and r 2 are the radii. 
If all the roots are positive and if « 4 is also positive, the curve 
consists of three portions, one of which extends to infinity and r x 
is its least distance from the origin, a second portion is limited by 
the circumferences of which r 2 and r 3 are the radii, and the third 
portion passes through the origin at each extremity, and r 4 is its 
greatest radius vector ; if « 4 is negative, the curve consists of two 
portions, one of which is limited by the circumferences of which r x 
and r 2 are the radii, and the other by the circumferences of which 
r z and r 4 are the radii. 

When # 4 is positive, the following notation may be adopted. 

9\ — r 3 = A tan £ e tan £ 1] , 
r i — r i = A cot £ £ tan £ / )] 1 , 
r 2 — r 4 = A tan £ e cot £ r\ , 
r 3 — r 4 = A cot £ e cot £ i]x , 



which give 



i=z £n — g 



tan 2 ^= tan?? 



tan //! ' 



— 396 — 



Or sin (r a — v) 

COS £ = 



sin (ft-)- J?) 

For the portion of the curve, which is contained between the cir- 
cumferences of which r 2 and r 3 are the radii, the notation 

r 2 sin 1 1] — I sin \ % , . 
r 3 cos ^i] = l cos h x , 



•j 



tan 2 (^-^)=: f n t^-^ 

v J tan -%?] {r. 2 — r) 

gives 

r sin ^ (^ -)- x) -(- sin \ (^ — x) sin d 

I sin i (?/! + tj) -\- sin |- (^ — ??) sin (9 ' 

The equation of the curve is, then, 

A (y — K)y/ci! 4 __ sintfa— ^) oj? ^ _|_ sirnfr sin ^ fa ■— x) qp I" sin 2 ! fa — x ) ^1 

2p{lcosi sin^fa — x) l "Tsinifa — x)sin-|-fa-j-x) l L sin 2 £fa-f- x)' J 

sin i (jj — x) y/ (sin^ cosecx) rj, y/ (1 — cos 2 itan 2 0) 

y/[sin a^ sin x — sin 2 i sin 2 ^ fa-|- x)] y/ [1 — sin 2 a sin 2 ^ fa-f- k) cosec^ cosec x] 

and the expression for the time may be obtained from this value 
of (cp — a ) by 'interchanging x and iy and multiplying by — f- 1 . 

The nature of the motion through the space exterior to the 
circumference of which r x is radius, and within the circumference 
of which r 4 is radius, may be derived from equations (396 5 _ 15 ) by 
changing r 3 to r x and r 2 to i\ and augmenting each of the angles 
r] and % by the magnitude n. 

When % is negative, the following notation may be adopted, 

r x — r s = A tan h £ tan i i] , 
7*2 — r± = A tan I e cot h t] , 
i\ — ?4 = A cot k s tan £ i^ , 
r 2, — r 3 = A cot h e cot -|" ^i , 



i = I %• 



The nature of the motion between the circumferences of which 



— 397 — 

r z and r 4 are the radii may, then, be expressed by the equations 
(396 5 _ 15 ), provided that r s is changed to r i} and r 2 to r 3 and the 
sign of a 4 is reversed. The character of the motion between the 
circumferences of which r x and r 2 are the radii, may be expressed 
by the same equations with the change of i\ to r 2 , and of r 2 to r 4 , 
the reversal of the sign of a 4 , and the increase of each of the 
angles t] and x by n. 

The elliptic integrals disappear when two of the roots are 
equal ; in this case, if ?\ denotes one of the equal roots, and if R 2 is 
the quotient of the division of the first number of (389 29 ) by 
(r — r x ) 2 so that the form of R 2 is 

R 2 = h 2 r 2 -J- hi r -\- h, 

the notation may be adopted 

2 h -\-h 1 r = 2R sj— h tan (6 \]—h) = — 2R\/h Tan (6 sjh) , 
J h -\-2h 2 r=2R <J—h 2 tan ($ 2 sJ—/ h ) = — 2R\Jh 2 Tan ($ 2 y//; 2 ), 

hri+ ?RR\ +h)r = - ^— 1 tan Wi V - 1) = Tan {R x 6,) y 
the equation of the curve is 

and the expression of the time is 

t — T = 6 2 -\-r 1 6 1 . 

When # 4 vanishes, if a z is positive, the notation may be adopted 

r x — r 2 = B 2 tan 2 \ e , 

r x — r 3 == B 2 cot 2 h e , 
r 2 cos 2 ie = l cos 2 h * , 
r 3 sin 2 1 e = I sin 2 h, x , 
i= in — e ; 



— 398 — 

and for the portion of the curve contained between the circum- 
ferences of which r 2 and r 3 are the radii, 

tan 2 (in — H) = tan 2 ie ^=^. 



The equation of this portion of the curve is, then, 

(w «) J ffo COS E r~f . , /-, COS 8 \ ,575 , 9 .. 

y ? „/ v , 3 = — g? . d 4- ( l ) QP,- ( — cos 2 x, 6) 

2 pi I cos i. cos x ' \ cos yJ v 7 

I cosx — cose ., [_!] // 1 -|- cos 2 * tan 2 \ 

~ sin x y/ (cos 2 £ — cos 2 '/.) y Vcos 2 i — sin 2 i cot 2 yJ ' 

and the expression of the time is obtained from this value of 
((jp — «) by interchanging £ and k and multiplying by -~. 

Upon the portion of the curve exterior to the circumference, 
of which r x is radius, the notation 

v y l-)-sino • J 

gives for the equation of the curve 

(y -«W«8 _ SM _ 2^ 2 ^ r_ fa- b* \* ^1 



2 j»f cos » r x — .B 2 (r x — B 2 ) {r x -\- B 2 ) 

B _ tan [-l] 2^ v /(?- 1 +r- 1 cos 2 ttan» 



V/^ (± ri B 2 cos 2 i — (^—B 2 ) 2 sin 2 i)] u "" ^ [ 4 r i ^' 2 cos2 * ~ ( r i — •fi 2 j a sin !, t] ' 
and for the expression of the time 

(^^sec^^+^^^-^S^-f^J^v/ll-fco^aan 2 ^) 

-\-2B 2 Tan [ - 1] y/(l + cos 2 i tan 2 d) . 

If « 3 is negative, the notation may be adopted 

r x — r 3 = B 2 tan 2 £e , 
r 2 — r, 3 = B 2 cot 2 £e ; 



— 399 — 

which, combined with that obtained from (397 27 _3i) by changing 
r z into r 2 and r 2 into r 1} gives (398 7 ) for the equation of the por- 
tion of the curve contained between the circumferences of which 
7\ and r 2 are the radii, while the expression of the time is derived 
by the process of (398 13 ). But with the notation obtained from 
(398 16 ) by changing r x into r 3 and reversing the sign of Z? 2 , the 
equations (398 19 ) and (398 25 ) become the equation of the curve 
and the expression of the time, upon the portion which is con- 
tained within the circumference of which r 3 is the radius. 

The form of the central force which corresponds to the dis- 
cussion of this section is 

692. If m is 2 in the first class of § 690, the expression of 
the central force is 

F=hrS + b 3 r*-\-b,r+ b - 3 , 

and the forms of the equation of the curve are obtained from 
those of §691 by changing r into r 2 , and (cp — a) into 2 (cp — a). 
But the expressions of the time require, moreover, the substitution 
for (390 26 ) of 

p — q K ' l ' 

for (391 14 ) of 

t — % = J ( Aa " s ) tan [ - 1] 4 ^ r2 + g « 

V \ai(16aa 4 — a^a^f ' ^[^(lGaa,, — Oia 3 )]' 

for (39223) of 

(t — r)^ ai =m^ 
for (393 9 ) of 

(*— r)v/— a 4 =£3s.0, 



— 400 



for (393 23 ) of 






(<-*)V 


1 T r 11 (^ cos a — ^) r2 "l~ (^ cos a ~ 


-r)r? 


for (394 12 ) of 


(t — 'r)^a 3 =i%&, 




for (394 25 ) of 


(t — r)s/ — a 3 =i^ i 6, 




for (396 18 ) with the form of (396 23 ) of 






{t — t) y/ # 4 = i cos & SFj $ , 




for (397 5 ) of 


(^ — t)\/( — a 4 )= i cos a 9^$, 




for (397 24 ) of 


^ — T = i^, 




for (398 n ) and (398 25 ) of 






(7 — t) « 3 = I COS « 9^ £ , 




and for (399 2 _ 6 ) 


of 


- 


r 


(£ — t) y/ — a 3 =i cos a 9^ 3 . 





693. In the special case of § 692, in which F is reduced to its 

F=h 



first term, so that 



'i' ? 



two of the roots of (389 29 ) are real and two are imaginary, so that 
the only portion of § 691, which is applicable to this case, is from 
(391 15 ) to (393 23 ). In this case, moreover, one of the real roots is 
positive and the other is negative if # 4 is positive, so that the curve 
extends to infinity ; but if # 4 is negative, both of the real roots must 
be positive, so that the circumferences which correspond to these 
roots are the limits of the curve, and S2 is negative and satisfies the 
condition 



— 401 — 

694. In the special case of § 692, in which F is reduced to 
its second term, so that 

F=b 3 r 3 , 
the equation (389 29 ) has no imaginary roots of r 2 when 



X"' 



When b 3 is positive, there is only one real root, so that the 
curve extends to infinity from the circumference, which is defined 
by this root. When b z is negative, all the roots must be real, and 
the two roots, which are positive, define the circumferences which 
limit the extent of the curve. 

695. If m is f in the first class of § 690, the expression of 
the central force is 

F= £ 4 r~i -j- b 2 r-$ -f h x r~* -\-l r~\ 

and the forms of the equation of the curve are obtained from 
those of §691 by changing r into r*, and cp — a into I ((p — a). 
But the formulae for the time are more complicated, although 
they are still reducible to elliptic integrals. If, indeed, 

2 

z = r», 

the expression for the time assumes the form 

i % _ P fz 2 

696. In the special case of § 695, in which F is reduced to 
its first term, so that 



F=b. 



i 



the conditions of the form of the curve are the same with those 

51 



— 402 — 

expressed in (400 23 _ 3 o), but instead of (400 3] ), the limitation of 
S2 when & 4 is negative is 

697. In the special case of § 695, in which F is reduced to 
its second term, so that 

F—hr-i, 

the equation (389 29 ) has no imaginary roots of \j r 2 , when 

hl<—±p\£2l 

In the special case, in which F is reduced to its third term, so that 

F=\r~i, 

the equation (389 29 ) has no imaginary roots, when 

In each of these cases, when S2 is negative, there is only 
one real positive root, so that the curve extends to infinity from 
the circumference which is defined by this root. When /2 is 
positive all the roots must be real, and the two roots, which are 
positive, define the circumferences, which limit the extent of the 
curve. 

698. If m is i in the first class of § 690, the expression of the 
central force is 

F=b s r~i -\- b 2 r -2 -j- ^ r~§ -\-br~ s , 

and the forms of the equation of the curve are obtained from those 
of § 691 by changing r into \J r and (<p — a) into 2 (<p — «)• But if 

2 = \]r, 



_ 403 — 
the expression of the time assumes the form 

X9 ~3 



-f- a 2 z 2 -\- a x z -\- a) ' 

699. In the special case of § 698, in which F is reduced to 
its first term, so that 

and in that, in which it is reduced to its third term, so that 

F=\r-§, 

two of the roots of (389 29 ) are real for \J r, and two are imaginary, 
so that the only portion of § 691, which is applicable to this case, 
is from (391 15 ) to (39324). I* 1 this case, moreover, one of the real 
roots is positive and the other is negative if S2 is negative, so that 
the curve extends to infinity ; but if S2 is positive, both of the real 
roots must be positive, so that the circumferences, which correspond 
to these roots, are the limits of the curve, and in the former of 
these cases b 3 is negative and 

— *>M?(6I2jJ), 

while in the latter case b x is negative and 

700. In the second class of §690, when m is unity, the equa- 
tion (378 22 ) of the curve assumes the form 

([) d I LI 

Jr)J {a i r i -\-a A r z -\-a 2 r' i -\-a l r-\-a)'> 

so that it can always be obtained from the expressions of (t — t) 
in § 692, by multiplying either of those expressions by 4j»;f. When, 
in this class, the curve terminates in the origin, it does not usually 



— 404 — 

pass through the condensed coil of § 691. The formula for the 
time is 

t- % =i t 

J r s/ («4 'f 4 + «3 r 3 + <h ? ' 2 + «i r + «) 

The form of the force, which corresponds to this case, is 

701. In the special case of § 700, in which F is reduced to its 
third term, so that 



F— h 



4 J 



one of the roots of (389 29 ) is zero, and the condition that all the 
roots are real is 

When S2 is negative, if h 1 is positive, the curve extends to 
infinity, in the space exterior to the circumference of which the 
positive root of (389 29 ) is the radius ; • but if b x is negative, the 
curve extends from the origin to infinity, if two of the roots are 
imaginary, but if all the roots are real, one portion is exterior to 
the circumference of which the greater positive root is radius and 
extends to infinity, while the other portion is contained within 
the circumference of which the smaller positive root is the radius, 
and this portion passes through the origin. When 12 is positive, b x 
is negative, and the curve passes through the origin, and is con- 
tained within the circumference of which the positive root of 
(389 29 ) is the radius. This case of force has been analyzed by 
Stader. 

702. In the special case of § 700, in which F is reduced to 
its last term, so that 

F=K. 



— 405 — 

all the roots of (389 29 ) are imaginary when S2 and b are both 
positive. When I2 is positive, therefore, b must be negative and 
the curve is contained within the circumference of which the pos- 
itive root of (389 29 ) is the radius. When I2 is negative, if b is 
positive the curve extends to infinity in the space exterior to the 
circumference of which the positive root is radius ; but if b is 
negative, the curve consists of two portions, one of which extends 
to infinity in the space exterior to the circumference of which the 
greater real root is radius, while the other portion passes through 
the origin and is contained within the circumference of which the 
smaller root is radius ; or it extends from the origin to infinity. 

703. When m is 2 in the second class of § 690, the form of 
the force is 

F= b±r -j- b 2 r~ 3 -\- b^' - ^ -\- br~ 7 , 

and the equation of the curve can be obtained in each case from 
that of §692, by multiplying {t — r) by 2p\, and changing t — % 
into (p — a, and r into r 2 . 
If 



. the formula for the time is 

Jz v ["■ 



4 z 4 -f- a 3 £ -\- a 2 z 2 -\- a x z -\- a] ' 
704. In the special case of 

r' ' 

there are two imaginary roots of r 2 when 

b^ 64p\ 2 ' 
When S2 is negative, if b is positive the curve extends to 



— 406 — 

infinity in the space exterior to the circumference of which the 
real root of (389 29 ) is the radius; but if b is negative, and if all 
the roots of (389 29 ) are also real and two of them positive, the 
curve consists of two portions, one of which extends to infinity in 
the space exterior to the circumference of which the greater posi- 
tive root is radius, while the other portion passes through the 
origin and is contained within the circumference of which the 
smaller positive root is radius ; but if neither of the roots is positive 
when b and I2 are both negative, the curve consists of a single 
portion which extends from the origin to infinity. When S2 is 
positive, b must be negative and the curve consists of a single 
portion which passes through the origin and is contained within 
the circumference of which the positive root is radius. This law 
of force has been analyzed by Stader. 

705. Another class of central force, in which the integration 
can be performed by elliptic integrals, corresponds to the form of 
the potential 

q b i r im -\-b s r 3m '-\-b 2 r 3m -{-b 1 r m -\-b 

~ r 2 (r'» + hf ~' 

in which m may be either 1 or 2. If, in these forms 

z = r m , 

Z 2 = di s 4 -|- a s z 3 -j- a 2 s 2 -J- % z -\- a 

= (2 b, r im -{-2b 3 r Sm -\-2b 2 r* m -f- 2 b x r -f 2 b) 

— (2£2 r* + 4: P t)(r™+hY, 

the equation of the curve assumes the form 
and the expression of the time is 



— 407 — 

706. The following graphic construction gives an easy geo- 
metrical process for tracing the various cases of limitation of the 
extent of the path described under the action of a central force, 
and especially for finding by inspection the effect of the values 
of S2 and jh upon the limits of the curve. If 

l 

r' 2 ' 

construct the curve of which the equation is 



y =a 



which may be called the potential curve, draw the straight line of 
which the equation is 

y=2p\x+£2, } 

and the points of intersection of the straight line with the potential 
curve give the values of x for the limits of the path of the body. 
The path corresponds to those portions of the potential curve 
which lie upon that side of the straight line, which is positive with 
respect to the direction of the axis of y. 

707. A term of £2 may be omitted in the preceding construc- 
tion which is inversely proportional to the square of the radius 
vector, and its negative may be combined with that term of the 
equation of the straight line which determines its direction. The 
omitted term corresponds to a term of the force which is inversely 
proportional to the cube of the radius vector, and which may be 
represented by (383 17 ) ; and the corresponding equation of the 
straight line is 

y=(2rf + *)* + fi . 

708. It is evident from the preceding construction, that if the 
potential curve has no point of contrary flexure, and if its convexity is turned 



— 408 — 

in the direction of the positive axis of y, the path of the bod// can only 
consist of a single portion which may have either an outer or an inner limit, 
or it may have neither or both. This case includes all forces of the form 

F=b 1 r™ + b ? , 

in which b x and m -\- 3 have the same sign. 

But if the potential curve has no point of contrary flexure, and if its 
convexity is turned in the direction of the negative axis of y, the path of 
the body may consist of a single portion ivhich has either an outer or an 
inner limit, or it may have neither, or it may consist of two separate por- 
tions of ivhich one has only an outer and the other only an inner limit. This 
case includes all forces of the form (408 5 ), in which b x and m -\- 3 
have different signs. 

709. Those portions of the potential curve, in which y and x 
simultaneously increase, correspond to the distances from the centre 
of action, at which the force is attractive, so that the convexity of 
the path of the body is turned away from the origin. The portions 
of the potential curve, in which y decreases with the increase of x, 
correspond to the distances from the centre of action, at which the 
force is repulsive, so that the convexity of the path of the body is 
turned towards the origin. Any point, therefore, at which the 
potential curve is parallel to the axis of x, and the ordinate is either 
a maximum or a minimum, corresponds to a distance from the 
origin, at which the central force changes from attraction to repul- 
sion, and the path of the body has a point of contrary flexure. 

710. If for an infinitesimal value of r denoted by i, 12 assumes 
the form 

£2 = ki n , 

the path of the body cannot pass through the origin if n-\-2 is 
positive or if k is negative, except in the former case, when p 1 van- 



— 409 — 

ishes and n is positive while S2 is negative, or n is negative while 
Jc is positive ; but if Jc is positive and n-\-2 negative, the external 
portion of the path passes through the origin, and after passing 
through the origin, the continuity of curvature is destroyed and the 
path becomes a straight line. 

711. If for an infinite value of r, denoted by the reciprocal 
of i, Si assumes the form (408 28 ), the path of the body cannot extend 
to infinity when n and Jc are both negative, or when n and 12 are 
both positive, or when n vanishes and 

but the external portion of the path extends to infinity when n is 
negative and Jc positive, or when n is positive and X2 negative, or 
when 11 vanishes and 

J2 <*. 

712. If a line is drawn parallel to the axis of x at the dis- 
tance I2 from this axis, and assumed as a new axis of x 1} and if 
y x and y % are the corresponding ordinates, respectively, of the 
straight line (407 u ) and of the potential curve, the value of the 
angle, which the path of the body makes with the radius vector, 
is given by the equation 

which admits of simple geometrical construction. If z 2 denotes the 
subtangent of the potential curve upon the axis of x 1} the projection 
of the radius of curvature of the path of the body upon its radius 
vector is 

pan : = -£!, 

which is constructed without difficulty. By the combination of 
these two constructions, the path of the body may be obtained with 
sufficient exactness for most purposes of general discussion. 

52 



— 410 — 

713. When the origin is infinitely remote from the body, 

the forces of § 676 are parallel, and the plane of motion is parallel to the 
direction of action, and the equation (378 5 ) gives, if the axis of z is 
supposed to have the same direction with the force, 

-2 cot* ^*, 



of which the integral is 



sin 2 f 



in which a is an arbitrary constant, which is always positive, and 
this is the equation of the 'path of the body referred to the same coordi- 
nates with those of § 571. 

714. In the case of a constant force, the preceding equation 
assumes the forms 



g(*— *o) = m 



a 

sin 2 _ 

2a 



^ g sin 3 1 ' 

so that, in this case, the path is a parabola. 

715. The velocity in the direction of the axis of x is 

v sin z = sin s ^ ( 2 £2 — 2 12 ) = y/ ( 2 a) , 

so that this velocity is constant, and 

x — x = y/(2a) [t — t). 



nates, is 



The equation of the curve, expressed in rectangular coordi- 



v-t 



— 411 — 

716. If a potential curve is constructed by the equation 
(407i O )> m which y may be changed into x, and S2 retained as a 
function of z, the limits of the path of the body are defined by 
the intersection of the potential curve with a line drawn parallel 
to the axis of z at the distance (S2 -\- a) from this axis. The por- 
tions of the potential curve which correspond to the path, lie in 
a positive direction from the intersecting line. 

717. If the force of § 713 has the form 

F=b 1 z-\-b, 
the equation of the path is 

b x z + b = s/ (b* + 2 h S2 + 2 h x a) sin [(x - x) y/ (- ^)] 
= ^(P + 2b 1 S2 +2b 1 a)Cos[(x-x ) s J^] 
= v/_ (P 4. l x S2 + 2 h a) Sin [(* - x) y/^] . 

718. If the force of § 713 has such a form that 

O \z-\-b 

ll o — j^ky, 

the notation 

b 1 = 2(k 1 + h)(£2 -{-a), 
5 = (F4-^)(i2 + a), 

gives, for the equation of the path, 

h—z _ • (x - x) y/ (.Q„+ a) — y/ (P -f- 2 h z — z*) 
y/ {& + %) — £ 1 + A 

which is easily transformed into the forms, which are appropriate 
when the radicals become imaginary. 

719. In the case of a surface of revolution, and a force which is 



— 412 — 

directed to a point upon the axis of revolution, the notation of ^ 576 
gives 

— ==w sin? = u 2 D s Z, 

A=uv sin ? = u 2 B t ", 

so that the elementary area described by the projection of the radius 
vector upon the plane of x y is constant. 
720. The notation of § 578 gives 

T) „_ t /2» 2 (-Q-^) [r 2 +(2V) 2 ] 
X V — y- 2v?(SZ — £„)— A 2 > 

n * — A J. r '+(^ r ) a 

? * m V 2 « (fl — .Q ) — ^t 2 ' 
and, in the case of parallel forces 



D z s = B z a s J 



2 u 2 (fl — .Q ) 
2ic 2 (tt — tt )—A 2 > 

AD, a 



^\2u 2 (P. — Sl )—A 2 Y 



721. Upon the surface of revolution which is defined by the 
equation 

uv = B, 

the path of the body makes a constant angle ivith the meridian curve. In 
the case of 

B = A, 

the path is perpendicular to the meridian, and is a circle of ivhich the 
plane is horizontal. 

Whatever is the value of B, for the point at which v vanishes u 
is infinite, while v is infinite when u vanishes.. 

Upon any other surface of revolution about the same axis, the in- 
clination of the path of the body to the meridian arc is the same ivith 



— 413 — 

the corresponding inclination upon the surface of equation (412 21 ) at the 
common circle of intersection of these two surfaces. Hence the limits of 
■ the path upon the given surface of revolution are its intersections with the 
surface of equation 

n v=.A, 

and the path extends over that portion of the given surface, which is ex- 
terior to this surface hy which the limits are defined. 

722. In the case of a heavy body the equation (412 2 i) be- 
comes 

M 2 2 =r — . 

2 9 

723. In the case of a heavy body upon a vertical right cone, if 
the body moves upon the inverted part of the cone, the path has an 
upper and a lower limit ; but if it moves upon the part, which is below 
the vertex, the path has an upper limit from which it extends doivmvards 
to infinity. In this case, if the notation of (341 13 ) and (341 16 ) is 
adopted, if two of the roots of the equation 

A 2 
r 2 (r — r ) = x — — 2 , 

are imaginary, which corresponds to 

(-fro) 3 <-TT " 



g sin 2 a cos a ' 



if h is the modulus and (i the argument of one of the imaginary 
roots, and if r x is the real root, the notation 

r 1 — hc^- 1 = B 2 c 2i - / -\ 

r 1 — hc-^- 1 = B z c- 2i ^-\ 

r — fi== B 2 tan 2 \ cp , 



— 414 — 
gives for the equation of the path upon the developed cone 

2i? tnn i-n 2^(cot'(p + cog'0 . 

V (2 r x — 2 B 2 cos 2 i) <J(2r 1 ^2B 2 cos 2 i) > 



and the position of the body at any instant is denned by the 
equation 

B{t — %)^{2gcota) = {r x + B 2 ) ® i <p — 2B 2 $ i <p 

-|-2_Z? 2 tan £<jp y/(l — sin 2 «sin 2 (j)). 

If all the roots of (413 20 ) are real, and denoted in the order of 
decreasing magnitude by r 1} r 2 , and r 3 , and if 

ri — r 2 = B 2 tan 2 (1, 
r x — r 3 = B 2 cot 2 [1 , 

i=2 /? + hn, 
r — r-i = /3 2 tan 2 ( I n — £<p), 

the equation of the path upon the lower portion of the devel- 
oped cone is 



j B n[ _u / ^^(l + cos'ttan'g,) 

~ r [^i(^i + ^ 2 ) 2 cos 2 z — 4r 2 B 2 ] V (r^B^cos't—^B 33 



and the position of the body at any instant is denned by the 
equation 

(t — r) cos t \J (2^ .cos a) = (^ -4- BJ cos 2 i &+ y — 2 B S f y 

-\- (jf-{-B sin (p) y/(l-4-cos 2 &tan 2 ^). 



— 415 — 

The equation of the path of the body upon the upper portion 
of the cone is determined by the combination of the equations 
(414 13 _ 16 ) with 



r x — r — B l 
Kin y sin i = 

sin 2 a B\J (g cos 



sin (j)sinj= j— ^r, , 

' r x — r-\-B z? 



B_ W _l] 4 / (l+cos'iWqOKrH-^Wi-^lP] 

N /[r 1 (r 1 + 5 2 ) ;J cos' 2 i— 4r 2 J B 2 ] V ±r x B* » 

and the position of the body at any instant is defined by the 
equation 

{t — t) cos i y/ (2g cos a) = — \j>-\- B) cos 2 / 3^ w 

■ sin i sin qp 



-j- 2 B § j 93 — 2 2? sin z cos yt/- 



-j- sin » sin 9 ' 



The path of the body upon the upper portion of the cone 
may be expressed in a somewhat more simple form by the equations 



snre = 



r = r 2 sin 2 (p -j- r 3 cos 2 9 , 

sin«V(2ycos«)(^-^ = r -^^^( r ^^), 

and the corresponding formula for the position of the body at any 
instant is 

In the special case, in which the roots r 2 and r 3 are equal, 
the path upon the upper portion is a horizontal circle, and the 
equation of the path upon the lower portion is 



(^_^ ) = ta^-Y'(-^-^)-v/3.tant-y(-l-^) i 



— 416 — 

while the position of the point at any instant is defined by the 
equation 

v/(2^cosa)(^-^) = 2 V /(r + ir ) + |- V /(-r )tan[-y(-i-^). 

724. In the case of a heavy body* upon the surface of a vertical 
paraboloid of revolution, of which the axis is directed downwards, the 
path has an upper limit, from which it proceeds doivnivards to infinity. 
If (336 25 ) is the equation of the paraboloid, and if z x and — q are 
the roots of the equation 

%pgz(z — s )=4 2 , 
the path of the body when 

is defined by the equations 

z — z x = {z x -\-p) tan 2 cp , 
q — p = [q -j- z x ) sin 2 i, 

and the position of the body at any instant is given by the equation 
% cos i(t — t)\J J_ =cos 2 i ( 3^ i (p — &i<p-\- y/(cos 2 /tan 2 cp -|- sin 2 /sin 2 y). 
But when 

the path is defined by the equations 

z — z x = {z x +p) cot 2 y, 
p — q = (z x -\-p) sin 2 i, 
«— A qp ( P_ w ) 



— 417 — 

and the position of the body at any instant is given by the equation 

{t — r)s/\_hg (si -\-p)] = % (p — &i ^ — si (cot 2 (p — sin 2 1 cos 2 y). 

In the especial case of 

p — q, 

the path of the body is the parabola, which is formed by the inter- 
section of the paraboloid with the vertical plane, of which the 
equation is 

u cos % =\J (4:/ -\-ul), 

and the position of the body at any instant is defined by the 
equation 

(t — r ) \J(p g) _, u 
V'(8^ + 2«iS) — tan - 

725. In the ease of a heavy body upon the surface of a vertical 
paraboloid, of which the axis is directed upward, the path has an upper 
and a lower limit. If p is negative, (336 25 ) is the equation of this 
paraboloid, and if — s x and — .« 2 are the roots of the equation 
(416^), they correspond to the limits of the path. The path of 
the body is defined by the formulae 

z = — z x cos 2 (f — s 2 sin 2 (p , 
(■?2 — p) sin 2 i = 2'2 — Zi , 

and the time is given by the equation 

,_, = v /!(i^)gr j(f . 
53 



418 



THE SPHERICAL PENDULUM. 



726. When the surface upon which the body moves is that 
of a sphere, the problem becomes that of the spherical pendulum. In 
this case, the path has an upper and a lower limit. If the centre 
of the sphere is the origin, if R is the radius of the sphere, the 
limits of the path correspond to the roots of the equation 

2g{R* — z 2 ){z — z,)—j?=0. 

If the roots of this equation are z 1} s 2 , and — p, and if the nota- 
tion is adopted 

z = z x cos 2 (p -4- z 2 sm2 9 ■> 
(p-\-z 1 )sm*i=z 1 — z 2 , 

the path is defined by the formula 
and the time by the formula 

727. From the equation (418 10 ), it is easily inferred that 

z x z 2 -\-R 2 = p{z l -\-Z 2 ), 
Z Q = Z x -J- Z 2 p , 

that the sum of z J and z 2 is always positive, and that p exceeds R. 

728. It is apparent from the inspection of (418 21 ) that, if the 
mutual ratios of R^ and the roots of (418 10 ) are unchanged, the 



— 419 — 

time of oscillation of the pendulum is proportional to the square root of 
its length. 

729. If the length of the pendulum and the sum of z x and p 
are given, it is evident from (418 21 ) that the time of oscillation 
increases with the increase of i, and is a minimum when i van- 
ishes, that is, when 

in which case the path of the pendulum is a horizontal circle. 
The time of oscillation in this case is 

„ r 2 n R 



\/[^0> + *i)]' 



The mutual relation of p and z ly which is here given by the equa- 
tion (418 26 ), is 

whence 

This value is a minimum, when 

z 1 \J 3 = R, 



in which case 

2R ___ /2z x 
9 



*=«v/#=V- 



which is, therefore, the greatest time of vibration when the path of 
the pendulum is a horizontal circle. 

It is easy to see that i cannot vanish for all values of the 
sum of p and s 1} but that its least value is determined by the 
equation 

sin 2 2 i = 4 — -. — , — v ,, 



— 420 — 
whenever 

It is also evident that the least value of the sum of p and % 
which corresponds to any assumed value of i is given by (4I9 30 ), 
so that for any value of i, the greatest time of vibration is 



2WIV / C- ± T^) 9, '<*»>= 



which increases with i, and is infinite ivhen i becomes a right angle. 

When i is an octant, the value of p -4- z\ in (419 30 ) is a maxi- 
mum, and the corresponding values of p -)- z x and T are 

p -f -0j = 2 R 

^=v/f^(*»)- 

730. In the discussion of the form of the path of the pendu- 
lum, it is convenient to adopt the notation 

In the case of (419 7 ), the equations of §726 and 727 give 

f- = 2 z, (/ — &) = { *~ A) \ 

JT 2nTt 

"'—)/<?* + &)' '- 
When 2j vanishes 

#=2tt, 

7T " 

and T is the time of a complete revolution. When 

v« = *> 



— 421 — 

and T is the time of a semi-revolution. The time of a complete 
revolution, when the pendulum moves in a horizontal circle is 



Tj=2n 



V 



~2 



so that it is proportional to the square root of the distance of the plane 
of revolution from the centre of the sphere. 

731. When the path of the pendulum deviates slightly from a 
horizontal circle, so that i is very small, the notation 

j? a -j- S 2 = 2 £g = 2 R cos <5 3 , 

gives 

*i — H = (j» + s's) i" = — f-r-^ i2 > 



2z„ 



2z 3 ' 



+ lt~ -I- d Zg .9 n 

— -j f cosz (p, 

732. When the path of the pendulum deviates slightly from 
a great circle, so that the sum of s 1 and g 2 is small, p is large and i 
is small, the formulae become, by neglecting the fourth and higher 
powers of i 

A 2 



(^-^[^-(^-^(I-^), 



z = i (*, - z 2 ) cos 2 y + - 4l ^y *-' * 2 , 

f 7. = 2 7T ; 



— 422 — 

so that the vibration corresponds to a complete revolution of the pen- 
dulum. 

733. When the pendulum passes very near the lower point 
of the sphere, so that % differs but little from R, the neglect of 
this difference and its higher powers gives 

2 2 = R cos 2 i, 
p=R-\-tan 2 i(R — g 1 ), 

^- k =4:R 2 (R — Sl )sm 2 i, 

z = R — 2 R sin 2 i sin 2 cp — [R — s x ) cos 2 c/> , 

« r .=» + [««o<».(*»)4-'*Ssr«,^»)] v /(2 :T ^)j 

so that the vibration corresponds to a little more than a semi-? •evolution 
of the pendulum. 

734. In the general case the vibration of the pendulum corresponds 
to an arc of revolution which exceeds a semi-revolution, but is less than 
an entire revolution. When the velocity at the highest point is quite 
small, the case of § 733 occurs, hut the arc of revolution, which cor- 
responds to a vibration, increases with the increase of velocity at 
the highest point. When the highest point is below the level of 
the centre of the sphere, the case of § 731 gives the highest limit 
of the velocity at this point ; but when the highest point is upon 
or above the level of the centre, the greatest velocity extends to 
infinity, which limit corresponds to the case of § 732. 

735. The azimuth of the pendulum at any instant, is derived 
from the equation of §726 in. a form suitable for computation by 
means of the following formulae ; 



z = Rcos 6, 

r, R (COS #, COS do -\- 1 ) 

p=Bseca=—± — - 1 , 2 l ' 

1 COS l -f- COS C7 2 



423 



A 1 m , » / ,, . N i? 5 sin 3 ft sin'-' ft, 

— =M S tan- « (cos d, 4- cos tU = — a - 

"2g v ' - ' cos d y -\- cos ft 



9 . 1-1- cos « cos ft, 
COS- 2 = =-J t-- , 

1 -j- COS « COS /7j 

sin -^ ^i 
Sm ^ = sin-I^' 

cos i ft., 

tan^^ 1+?0S,?lC0sf 



sin # 2 cos ft 

cos ft -)- cos ft. 



tan ^x = cos a cos d 2 tan fi = 



sin (?! 



cos ,u = cos ^ 2 cos ^ , 
co & = h Ti — i, 

n \/ (cot 2 cr — sin 2 i cos 2 ai) 

tan /.! = s-v * . , *' , 

cos i cos jw x sin ft, 

, sin // tan ft, tan i ft, tan P. 

tan l 2 = — £— — -- — 2 — , 

tan ft l cos fi x 

, , tan 2. 

3 tan 2 //(l + cos 2 i tan 2 9) ' 

, , tan 2 i cos 2 9 cos ft 2 tan ju tan | 6i tan X 

tan A4 j — jz - — . „ . — r-5 — , 

COS [A COS 7] COS //j -(- (1 COS fl COS 7] COS ?/,) Sill" I Sill" (f 

TT cos i sin u tan ft r. Tf) / cos 2 i cos 2 u tan 2 ft, \ ~ 1 
^ = " Ian ir L ®* ( ta^ > V ~ ^ 9»J 

-J- cos i cos ^ tan <3 2 ^ 9 -|- ^-1 — ^2 -f~ ^3 — ^t> 
and the arc of revolution for the complete vibration is 

**£=«+ [#<(**)_*< (i TT)] ^ co ^_^(i *) g cQi u 

-J- cos i cos (it tan $ 2 9*,- (i tt) ■ 

These formulae do not appear to differ from those of Guder- 
mann, although the reduction is more extended. They give with 
equal facility the area of spherical surface which is described by the 
arc of a great circle, which joins the extremity of the pendulum to 
the lower point of the sphere. 



— 424 — 



MOTION OP A FREE POINT. 

736. When a material point is unconstrained by any condi- 
tion, and is free to obey the action of any force whatever, its 
motion in any direction is simply denned by the equation 

D* x = D x tt . 

737. If the coordinates are assumed to be of the partial polar 
form in which 

z = the distance from the plane of xy , 

o = the distance from the axis of z, 

(p = the inclination of o to the axis of x, 

the value of T (162 28 ) is, for the unit of mass, 
The corresponding values of to (165 4 ) are 

(D = Z', 

«>! = </, 

co 2 = o 2 <jp' ; 

so that cd 2 is the double of the projection of the instantaneous area, which 
is described by the radius vector of the point, upon the plane o/xy. 
The equation (166 2 ) gives, then, 

It is apparent from (39 ]0 ) that the second member of the last equation 



— 425 — 

is the moment, with reference to the axis of z, of all the forces zvhich 
act upon the point. 

738. If the forces are proportional to the distances from the centres 
from which they emanate, the hody moves as if it were under the influence 
of a single force, acting by the same law with an intensity equal to the sum 
of the intensities of the given forces, and emanating from a centre zvhich is the 
centre of gravity of the given centres regarded as masses proportional to the 
intensities of their action. For, if the notation employed in § 128 is 
adopted, and if m denotes the sum of the intensities of action, the 
value of the potential is 



a 



= i mr 2 -\- I o 2 = £ m r 2 -\- K, 



in which Zisa constant and can be absorbed into the constant H 
with which S2 is connected in the equations of motion. 

It follows from § 685, that the path of the body is, in this case, 
a conic section, of zvhich the centre of gravity is the centre. 

739. If all the forces are directed towards a fixed line, the area 
described by the projection of the radius vector upon a plane perpendic- 
ular to the fixed line is proportional to the time of description. For the 
instantaneous area is in this case constant by the equation (424 29 ), in 
which the fixed line may be assumed for the axis of z, so that the 
second member shall vanish. 

740. In the example of the preceding section, a peculiar sys- 
tem of coordinates may be advantageously adopted. This system 
consists of the sum of the distances from two fixed points of the 
given line, the difference of these distances, and the angle which 
is made by a plane passing through a fixed line, with a fixed plane 
which includes this line. If, then, 

2p = the sum of the distances of the body from the two fixed 
points, 

54 



— 426 — 

2 q — the difference of these distances, 
(p = the angle which the plane including the body and the 
fixed line makes with the fixed plane, 
2 a = the distance of the fixed points from each other, 
2 y = the angle which the two lines, which are drawn from 
the body to the fixed points, make with each other, 
k = the perpendicular drawn from the body to the fixed lines, 
pl=p 2 — a 2 , 

the values of /c,y, and T are 



a ' 



tan w — — , 
T Px 



,2 ,2 



2cos 2 ii; ' 2sm 2 ip ' ' yj - ' ■ LJ \pi ' q{/ ' 2 a 2 

The corresponding differential equations derived from Le- 
grange's canonical forms (164 12 ) are 

d, a = (i +$?-, |- ? -#+ i -^"'"' + " '** 



m 

2qlpp'q'-\-qp\q 
9l 



The integral of (426 20 ) is 

k 2 (p 1 = B, 

in which B is arbitrary, and this equation expresses the proposition 
of § 739, and gives 



9 — & 2 ~M' 



— 427 — 

A second integral of these equations, corresponding to the 
principle of living forces, is 

The sum of the equations obtained by multiplying (42622) by 
p\p r , (426.*) by (—fig) and (42T 3 ) by 2 (pp' + qq') is 

* A [(P 2 - <?) (/- /)] - D t [(/ + f - « 2 ) {£2 + //)] 
— (fp'D p S2-{-p*q'l) q £2). 

This equation is integrable, whenever 12 satisfies the condition 

2qD p n — 2pD (1 I2=(f — f)I)l q £2, 

which, by the substitution of 

_ l l 

p-\-q> J — p + q' 

may be transformed into 

If, then, 

I2j = a 2 Z>, 12 — / D y £2 — (x—y) £2 

= -(^.11—^1) = - ^-?^ 

the equation (427 i7 ) or (427 n ) becomes 

0-^^^-fF 2 ^^— (tf-fy)^ 

= -(2> > i2+4^-,) = - A[(f ?-^ )fl] - 

\ P I p2 g2/ pi g2 

If, then, P and Q are arbitrary functions of p and ^ respective- 
ly, the general value of £2 is 

f — 9 2 



— 428 — 

and, if 

P\=2H{p i — <)-f-2(P-f C)(f— a 2 ) — a 2 B\ 

Ql = 2R(q i — a i ) J r 2(Q—C) ^ — f) — a*B\ 
in which C is an arbitrary constant, the integral of (427 8 ) is 

HP 2 -??{/-/) = PI- Qh 

while (427 3 ) may be written in the form 

(/ - ff (<fi/+pl /) = & p\ +A Ql 

It is easy to deduce from these equations 

(f-hp'=Px 

(p*-f)</=Q 1 , 

J p Pi J q Ql 

d p -» 1 U q V 1 

tp = a*B f-i--fa 2 j? fJL. 

This solution is published by me in Gould's Astronomical Journal. 

741. It is evident from the linear form of the equations 
with reference to £2, that all special values of 12 may be combined 
into a more general value by addition or subtraction. 

742. The integrals in the values (428 15 _ 20 ) assume the elliptic 
form, when P and Q have the forms 

P = A + A lP + A i f + ^ p + . p ^ a , 

Q = B + B ll ±B 2 ? + a -^- + £ i , 

and it is apparent that, in the expressions of the integrals, the con- 



— 429 — 

stants, Aq, A 2 , B and B 2 can be combined with II, C, and B. The 
elliptic integrals become circular, when 

&=—A 2 =B 2 , 

A 1 = B 1 =0, 

as well as in other cases which do not seem to be of especial 
interest. 

743. When P and Q have the forms 

P = A,p, Q = B,q, 
the value of the potential is 

n- A + A ■ A-B, 

so that, in this case, the forces are equivalent to two emanating from the 
fixed points with the same law of force as that of gravitation, which case 
has been integrated by Euler, Lagrange, and Jacobi, and the forms 
of Lagrange's integrals are identical with those of (428 1& _ 2 o)- 

744. When P and Q have the forms 

P = Af, Q = —Aq\ 
the value of the potential is 

so that, in this case, the force is equivalent to a single force emanating 
from a point which is midway between the tivo fixed points, and the law 
of force is proportional to the distance from the centre of force, and this 
case is integrated by Euler and Lagrange. 

745. When P and Q have the form 

n — A — A Q— A — A 

pi jr — -a 11 ^ q{ or — q 27 



— 430 — 
the value of the potential is 

pi q'l a 2 h % ' 

so that, in this case, the force is equivalent to a force which emanates 
from an infinite axis of uniform extent, and is inversely proportional to 
the cube of the distance from the axis. 

746. When the curve is given upon which a material point 
moves freely, the law of the fixed force is restricted within certain 
limits which it may be interesting to investigate. The geometrical 
conditions of the force are simply that it must be directed in the 
osculating plane of the curve, and the normal force must be equal 
to the centrifugal force of the body. 

By the adoption of the notation 

il x = S2 -f H= T, 

the equality of the force in the direction of the normal N, or of 
the radius of curvature q to the centrifugal force is expressed by 
the equation - 

D N £2 1 = ^. 

747. Since the preceding equation is linear, all the special values 
of I2 X by which it is satisfied, may be combined into a neiv value by ad- 
dition or subtraction. Previously to this addition, each value of S2 1 may 
be multiplied by a factor, which may represent the mass of the body, 
and if the factor is denoted by m, the value of m X2 X will correspond 
to the whole force acting upon the mass, and it is, then, evident 
that, if M denotes the mass upon which the combined forces act, 
and V its velocity, the combined power is 

M V 2 =Z(mv 2 ), 
which expresses a condition identical with the theorem of Bonnet. 



— 431 — 

748. If a special value of I2 2 is represented by S2 , and if i2 2 
satisfies the equation 

B N n 2 =o, 

so that it is the potential of a force, to the level surfaces of which 
the given curve is a perpendicular trajectory, the complete value of 

n 1 = n f(£2 2 ), 



£2i is 



in which f is an arbitrary function. It is apparent, then, that £2 X has 
an endless variety of possible forms in every special case. But 
each form corresponds to an arbitrary value of one of the constants 
of the given curve, or of some combination of those constants. 

749. If the given curve is the parabola, of which the equa- 
tion is 

{l/—//o) 2 = 2p {% — %*), 

the values, which correspond to the arbitrary value of x , are 



X 



•*2a=.log(y.— ^o) + - 

n = 1 

while those, which correspond to the arbitrary value of y , are 



12 



■t+ts/^ 



£2 =2(% — x () )-\-p; 

and it is interesting to observe that when, in this case, the arbitrary 
function of S2 2 is assumed to be constant, the value of the force is 
independent of % and p as well as of y . 

The values, which correspond to the arbitrary value of p, are 



— 432 — 

750. If the given curve is the conic section, of which the 

equation is 

P 

ecos((p — cp ) = - — 1, 

the values, which correspond to arbitrary values of <p are 
n —ml* Ptan r ~ lir + P 1 sjn r -* 2i> - (1 ~~ fl *> r 

S2 =l-e*- 2 -f, 

in which 

E 2 = e 2 r i — (Pr — r 2 f. 

"When the arbitrary function of I2 2 is assumed to be constant, 
the force is independent of e and P as well as of <jp , and its law is 
identical with that of gravitation. 

751. If the given curve is the cycloid determined by the 
equations 

^—#0 = ^(1 — cos &)> 
x — x = E (& — sin G) ; 

the values which correspond to arbitrary values of x are 

£2 2 = x-\- E(6 -j-sin£), 
l 



S2 = 



y—yo' 



in which 6 is to be regarded as the function of y, which is deter- 
mined by (432 18 ). 

752. If the given curve is a circle of which the centre is the ori- 
gin zvhile the radius is arbitrary, the potential of the force is an arbitrary 
homogeneous function of the reciprocals of x and y, which is of the 
second degree. 



— 433 — 

This peculiar result is the more worthy of attention because 
it can be extended to the sphere, so that the potential of a force by 
zvhich a body may move upon a sphere of a given centre but of an ar- 
bitrary radius, is likewise an arbitrary homogeneous function of the second 
degree of the reciprocals of the rectangular coordinates, of which the centre 
of the sphere is the origin. 

These problems are fruitful of new subjects of interesting geo- 
metric speculation. 



CHAPTER XII. 

MOTION OF ROTATION. 

753. If the coordinates of the points of a system are the 
partial polar coordinates of § 737, and if y is supposed to refer to 
some point of the system, that is, to an axis connected with the 
system, from which the corresponding angles 6 are measured, so 
that the value of q> is 

SP = 9>o + d, 

that of T becomes 

Hence the equation (164 12 ) gives 

the second member of which is the derivative of double the sum of 
the products obtained by multiplying each element of mass by the 

55 



— 434 — 

area described by the projection of the radius vector upon the 
plane perpendicular to the axis of rotation. If this area is desig- 
nated as the rotation-area for the axis, it follows from (39 23 ) that the 
derivative of the rotation-area for the axis is equal to the sum of the 
moments of the forces ivith reference to that axis. It is obvious that 
the mutual actions of the system may be neglected in obtaining 
the sum of the moments. 

If, then, all the external forces ivhich act upon a system are directed 
towards an axis, the rotation-area for that axis will be described with a uni- 
form motion, which is the principle of the Conservation of Areas. 

754. The rotation-area for an axis may be exhibited geomet- 
rically by a portion of the axis which is taken proportional to the 
area, and it is evident from the theory of projections that rotation- 
areas for different axes may be combined by the same laws with 
which forces applied to a point, and rotations are combined, so that 
there is a corresponding parallelopiped of rotation-areas. There is, then, 
for every system an axis of resultant rotation-area, ivith reference to ivhich 
the rotation is a maximum, and the rotation-area for any other axis is the 
corresponding projection of the resultant rotation-area. The rotation-area 
vanishes, therefore, for an axis ivhich is perpendicular to the axis of 
resultant rotation-area. 



ROTATION OF A SOLID BODY. 

755. In the rotation of a solid body, the axis of rotation does 
not usually coincide with that of resultant or maximum rotation- 
area ; and the relations of these two axes is of fundamental impor- 
tance in the investigation of the rotation. The determination of 
these relations depends directly upon the moment of inertia. The 
moment of inertia of a body or system of bodies upon an axis is the sum 



— 435 — 

of the products obtained by multiplying each element of mass by the square 
of its distance from the axis. 

The distorting moment ivith reference to two rectangidar axes is the 
sum of the products obtained by multiplying each element of mass by the 
products of its distances from the two corresponding coordinate planes. 

Let then 

m = the mass of the body, 

Q p = the distance of the element d m from the axis of p , which 
passes through the origin, 

- Ip= the reciprocal of the moment of inertia for the axis of p, 

m J p = the distorting motion of inertia for the two axes which form 
a rectangular system with the axis of p , 

which gives 

-*p *Jm 

If, then, cp g is the angle which the axis of p makes with the direc- 
tion of q, the moment divided by the mass, becomes 

=^j[[^^ 2 -(^^ cos ^)) 2 ]' 

= S x (pj^ — 2J X cos (p y cos <p z ) . 

If I p is set off upon the axis from the origin, its extremity 
lies upon a finite surface of the second degree, which is, therefore, 
an ellipsoid, and may be called the inverse ellipsoid of inertia. If the 
axes of this ellipsoid are assumed for the axes of coordinates, the 
values of J must vanish for each of these axes, that is, there is no 



— 436 — 

distorting inertia for these axes which may he called the principal axes of 
inertia. 

756. When a body rotates about an axis, the rotation-area 
for an axis, which is perpendicular to that of rotation, is obviously 
proportional to the distorting inertia for these two axes. There is, 
therefore, no rotation-area for a principal axis of inertia proceeding from 
rotation about either of the other tivo axes of inertia. 

757. If Qp is the velocity of rotation about the axis of p, the 
corresponding velocity of rotation about the principal axis of x is 

& f =ff p coa<p xf 

and the corresponding rotation-area is 

m dp cos cp x 



B 



) 



the cosines of the angles, which the axis of resultant rotation-area 
makes with the principal axes, are then proportional to 



cosg^ cos cp v , cos go 



and 






so that this axis coincides with the perpendicular to the tangent 
plane of the ellipsoid which is drawn at the extremity of the axis 
of rotation. The plane of maximum rotation-area is, therefore, conjugate 
to the diameter of the ellipsoid ivhich is the axis of rotation, which theorem 
is given by Poinsot. 

758. If the reciprocal of the perpendicular let fall from 
the origin upon the tangent plane of the ellipsoid is set off upon 
the perpendicular, its extremity lies upon a second ellipsoid, which 
may be called the ellipsoid of inertia, and of which the principal axes 
are the reciprocals of the principal axes of the ellipsoid of § 755, and are 
proportional to the square root of the principal moments of inertia. 

759. It is apparent that the tangent plane to the ellipsoid of inertia 



— 437 — 

which is draivn at the extremity of the axis of maximum rotation-area is 
perpendicular to the axis of rotation. 

It is also evident, that the axis of rotation is one of the principal 
axes of the section of the inverse ellipsoid of inertia, which is made by a 
plane passing through the axis of inertia and perpendicular to the common 
plane of the axis of rotation and of maximum rotation-area, while the latter 
axis is one of the principal axes of the section of the ellipsoid of inertia, 
which is made by a passing plane through this axis perpendicidar to this 
same common plane. 

760. Although the fixed axes of coordinates may be assumed 
at any instant to coincide with the principal axes of inertia, the axes 
of inertia are nevertheless in constant motion from the fixed axes, 
and at the end of the instant dt, after coincidence, the axes of rota- 
tion, which coincided at the beginning of the instant with the fixed 
axes of y and z, will not remain perpendicular to the fixed axis 
of x, but will deviate from perpendicularity by the respective angles 

fr z dt and — &' v dt. 

The rate of increase of the rotation-area for the fixed axis of x, 
which arises from the external forces is, therefore, 



m B °* n —jf — V* 6 * \E~~W 



which represents the well-known equations given by Euler for the rota- 
tion of a solid body. 

If the rotation-area for the axis of p is denoted by m A' p , the 
preceding equation may assume the form 



lD e n = D t A x -A' y A' z {I?-l!). 



761. If the equation (43722) is multiplied by 2 d x and added 



— 438 — 

to the corresponding products for the other axes, the integral of 
the sum is 

m — *I>> 

which is simply the equation of living forces. If p is the semidiame- 
ter of the inverse ellipsoid of inertia, about which the solid is 
revolving at the instant, the preceding equation may be reduced to 

f>J r H _ .,2 cos 2 ^ _^ 2 



ROTATION OF A SOLID BODY WHICH IS SUBJECT TO NO EXTERNAL ACTION. 

762. If a solid body is subject to no external force, the centre 
of gravity may be assumed for the origin. In this case the first 
member of (437 22 ) or (437 29 ) vanishes, and the equation (438 9 ) 
becomes 

2 

p 2 m 

or 

&' p = hp, 

so that the velocity of rotation is proportional to the diameter of the inverse 
ellipsoid tvhich is the axis of instantaneous rotation, which is given by 
Poinsot. 

763. It follows from §§ 757 and 762, that, if q is the perpen- 
dicular let fall upon the tangent plane which is drawn to the 
inverse ellipsoid at the extremity of the axis of rotation, q is the 
axis of maximum rotation-area, which is invariable when there is 
no external force, and that 

ft' h 2 



— 439 — 
&' q = & p cos q> p = hp cos % = hq= -; 

-°-q 

so that the velocity of rotation about the axis of maximum rotation-area, 
as ivell as the distance of the tangent plane which is draivn to the inverse 
ellipsoid of inertia at the extremity of the axis of rotation are invariable 
during the motion of the solid, which are propositions given by Poinsot. 
They might have been deduced with facility from the geometrical 
theorem of § 759, without the aid of the equation of living forces, 
which might on the contrary have been derived, in the present case, 
as an inference from these theorems, and this was the elegant pro- 
cess of Poinsot. 

If the solid body has no translation, the inverse ellipsoid re- 
mains constantly tangent to the same plane which is that of max- 
imum rotation-area, and which touches the ellipsoid at the extremity 
of the axis of rotation. It is apparent, then, that in the motion of 
the solid, the ellipsoid rolls upon the fixed plane of maximum rotation-area, 
without any sliding ; which is Poinsot's mode of conceiving this 
motion. 

764. The instantaneous axis moves within the body in such a way 
as to describe the surface of the cone of the second degree, of which the 
equation is 



-4f(W)]=°- 



The base of this cone is an ellipse perpendicular to the greatest axis of 
the inverse ellipsoid when q is larger than the middle axis, or perpen- 
dicular to the least axis, when q is less than the middle axis ; and in 
either case the centre of the ellipse is upon the axis to which it is per- 
pendicular. 

When q is equal to either the greatest or the least axis, this 
axis becomes the permanent axis of rotation ; but when q is equal 



— 440 — 

to the middle axis, the cone is reduced to a plane which corresponds 
to one of the plane circular sections of the ellipsoid of inertia. 

The axis of maximum rotation-area describes within the body 
the cone of the second degree of which the equation is 

2.1(2 — f)*?\ = Q. 

The common plane of the instantaneous axis of rotation and of the 
axis of maximum rotation-area is obviously normal to the surface of the 
cone described in the body by the axis of maximum rotation-area, which 
defines the relative position of these two axes at each instant. 

765. The position of the axis of maximum rotation-area is 
fixed in space, and, therefore, the path of the instantaneous axis 
of rotation in space is determined by the preceding property, and a 
distinct geometrical idea of the cone described by the instantaneous 
axis in space, is obtained by conceiving the cone described in the 
body by the axis of maximum rotation-area to be compressed into 
a line carrying with it the cone described by the instantaneous axis, 
in such a way as not to change the relative inclination of the two 
axes or the surface of the cone of the instantaneous axis. 

The algebraic definition of the cone of the instantaneous axis 
in space is obtained by assuming the axes of the inverse ellipsoid 
to be arranged in the order of magnitude as x, y, s, in which 
the cone of the axis of rotation has the axis of x a, 9 its central axis, 
and adopting the notation 

cos JE!,. 

jC=j?+j7_^=/_^(i-^)(i-5) j 

and similar equations for the other axes, in which it is unimportant 
that the angles E x may be imaginary, but it should be observed 











!? + %- 




= <?- 


-ti 1 -? 



— 441 — 

that My is the largest of the quantities, M x , M y , and M x \ the fol- 
lowing notation is also to be adopted. 

sin 2 E = — sin 2 E x sin 2 E y sin 2 E z , 

* = (l_|)(i-5) (l _*) 

=v/[(i-f)(i-f)(i-f)]. 



COS7J 


— M,* 




COS ?]' 


M z 
— MJ 




sin i 


sin ?/ 
sin tf ' 




sin 9 


JW" 


-^) 


J/j, sin 


* ' 



sin 8 = sin e sin (p ; 

which give for each axis, if x y y, z denote the extremity of the in- 
stantaneous axis upon the surface of the inverse ellipsoid, 

I 2 x 2 sin 2 E = I 2 (f — M 2 ) sin 2 E x , 

Dx _ p Dp 

~x~~ p* — Mi 1 

r2 ,, 9 . a n p 2 I 2 sin 2 E x D p 2 
I 2 Dx 2 sm 2 E x ■=z l —^— T . hr^~- 

p 2 — M£ 

If, then, y is the angle which the plane of p and q makes 
with a fixed plane passing through q, the cone of the instantaneous 
axis in space is defined by the equation 

/sin 2 9 ,i> ¥ 2 + ^-=i>^ 2 +i>/ + i>, 2 , 
or 

T) .. __ qpD < j,p(p 2 — q 2 +q 2 N) 

iV - - (p 2 - q 2 ) ^(pt-M 2 ) (M 2 -p 2 ) (p 2 -MZ)-\ 

q sec d - (q 2 -j- M 2 — M 2 sin 2 t\ sin 2 cp) M y sin tf < 
~ M y sin tf i q 3 J¥ seed ' 

56 



_ 442 — 
which gives by the use of elliptic integrals 

766. The velocity with which the instantaneous axis moves 
in the body is readily obtained from the equations of the preced- 
ing section, which give in combination with (43722) 

sin 2 E P 

p D t p = 2 x (zD t x) = —hx?/z2: x — *-je 

■*-z 

hxy 



^ 2 S x (I x % 2 sin 2 ^) = h tt(p 2 —M?) ( Mf-p*) {f-Ml)-\ ; 



whence 

hDJ = 



$ M y smr} n 

and by elliptic integrals 

M y sin if h (t — r ) = &t (p . 

767. In the especial case of 

the axis of maximum rotation-area describes one of the circular sections of 
the ellipsoid of inertia, and the equations of § 765 become 

M x = M z = I y , 
j 

COS fj = COS V] = -^ , 

My 

i= 2 n , 
hl y (t — <v) = y, 
s/ (M 2 —p 2 ) = M qj sin rj Tan [h (t — t) Jf y sin 17] 

= \J{M 2 — 1 2 sec 2 9 2 ) = J^ sin 17 Tan ( ^^"^ . 



— 443 — 
The greatest value of p is, then, M v , which corresponds to 

t — % = ; 

and its least value is I y , which corresponds to 

t — r = ± oo . 

If the axis of rotation, therefore, coincides with the mean axis of the 
ellipsoid of inertia at the commencement of the motion, its position will be 
permanent in the ellipsoid, although it is affected with an element of insta- 
bility ; but, in all other instances of the present case, the axis of rotation 
describes the spiral of which (442 30 ) is the equation, and is constantly ap- 
proaching the mean axis at such a diminishing rate of velocity that it never 
reaches this axis. 

768. When the ellipsoid of inertia is one of revolution, the 
cones, described by the instantaneous axis in the body and in space, 
are both of them cones of revolution, so that the simplicity of 
this case requires no further illustration ; but it may be ob- 
served, that, when the ellipsoid is oblate, the moving cone rolls 
externally upon the stationary, but internally when the ellipsoid 
is prolate. 

769. This analysis, which is substantially the same with one 
of the forms of Poinsot, comprehends the principal conclusions of 
Euler, Lagrange, and Laplace, and may be extended to the case in 
which the origin is any fixed point of the solid. 



THE GYROSCOPE AND THE TOJP. 

770. When the solid is subject to any accelerated force, and 
its gyration is about a fixed point, which may be assumed as the 
origin, and when the ellipsoid of inertia with reference to this 



— 444 — 

point is an ellipsoid of revolution about the axis of z, the corre- 
sponding Eulerian equation is 

771. If the body is also symmetrical about the axis of z, the 
preceding equation becomes 

# z = », 

so that the rotation about the axis of z is uniform. 

772. If the force is that of gravitation, the problem becomes 
that of the gyroscope. If g is the direction of gravitation, h that 
of a horizontal axis, which is perpendicular to the axis of z, and 
has that direction about which the rotation from g to z is posi- 
tive, if 



j -> 



l= s 



«j 



h 
x) 



if I is the distance of the centre of gravity from the origin, l s the 
projection of I upon g, and if the gyration of the body is resolved 
into the three rotations, %' about the axis of z, £' about the axis 
of h, and ifj' about the axis of g, the rotation about the prin- 
cipal axes are 

G' z = i// cos £ -f- /, 

^ = w' cos I + £' cos % > 
& y = i// cos § — £' sin / ■ 

These equations give 

6' x cos I -f d; cos | = v'' sin2 1 = ¥ (i — |) • 



— 445 — 

The area about the axis of g is evidently described uniformly 
by the principles of § 753, so that if 

and if 4 is a constant, we have 
l z (6' x cos | -f tf y cos f ) -f a 2 1 6' z cos f = (P — £) i// -f- » « 2 4 = n a 2 4 . 
The equation (438 3 ) gives, in the present case, 

{P - id 2 /+ p r s 2 = h 2 {i 2 - id (4 - 4) , 

provided the constants 7i and 4 are determined by the equations 

h 2 = 2gl 2 I 2 , 

The elimination of if' from the equations (445 7 ) and (445 10 ) 
gives 

p f g = W {I 2 - 1 2 ) (4 - 4) - n 2 a* (4 - 4) 2 . 

773. The limiting values of l s correspond to the vanishing 
values of l g , and, therefore, reduce the second member of (445 19 ) to 
zero. If these values are denoted by l l7 l 2 , and — p, it is evident, 
from the form of the equation, that p is greater than I, while 4 and 
4 are included between — <?and-|-/. The equations for the spher- 
ical pendulum of §§ 726 and 727 may be directly applied to the 
gyroscope by changing z into l g and z 0} g 1} and z 2 m to 4? 4? an d 4? 
which give, by (41824_2 7 ), 

4 = 4 ~j~T ? 



— 446 — 



nr a 



With this notation and the equations derived from (418 13 _ 14 ), the 
expression of the time is 

and the equation of the path described by the axis of this body 
in space is 

which admits of reduction by the process of § 735. 

774. When the velocity n vanishes, the gyroscope is re- 
duced to a case of the spherical pendulum of which the length is 

775. When the two roots l x and / 2 are equal, the path of the 
gyroscope is a horizontal circle. The values of l s , and of the velocity 
of rotation can be determined for this case by the equations 

7 7 _ (?—%) (h-Q 
h h— p^_ 2 l 1 l i —3l*> 

a i n 2 _ (P-\-2l 1 l i — 3Zf) 2 

~W (p-ro^-i,) ■ 
The denominator of the value of (4 — 4) can be written in the form 

p + 44_3J? = 3(4 — hHh + k), 

in which 4 and 4 are positive quantities. If, then, l t is greater 
than 4, </ ^ s positive; but when 4 is contained between l h and l 6 , y' 
is negative ; and l x can never be contained between — 4 an d — I- 



— 447 — 

776. When the values of 4 and 4 are equal, if' vanishes at 
the same time with £', and we shall also find 

1 3 =z 4 == 4 > 
and the equation for determining 4 and jo is 

/* 2 (7 2 — 4 2 ) = «V(4 — 4). 

This is, approximately, the ordinary case of the gyroscope, and it is 
evident that in this case the values of 4 and 4 cannot be equal, 
unless 

h=.h 

so that the centre of the gyroscope cannot under these circumstances de- 
scribe a horizontal circle, which coincides with the conclusion of Major 
jft-V" -=& G. Barnard. 

If, however, in this case, n is very large, it is obvious that the differ- 
ence between 1 2 and \ is quite small, for this difference is 

h — h — n * a t ' 

which is also one of the results obtained by Major Barnard. 

777. When 4 is algebraically greater than 4? it is also alge- 
braically included between 4 and 4> so that if' is positive at the 
upper limit, and negative at the lower limit. But when l s is alge- 
braically smaller than 4? it is also algebraically smaller than either 
4 or 4 5 so that, in this case, if' is always negative. 

778. When 4 is equal to /, it is also equal to l 3 , that is, 

The velocity of rotation which corresponds to this case is deter- 
mined by the equation 

h* - I-I 2 p+l ' 



— 448 — 
which gives 



V 



— '"•" l-k 



■ ■_ {i—k y 

Bini — 2i{i-ky 

779. When 4 is equal to — I, it is also equal to l a , that is, 

l\ = — I = / 3 . 

In this case / 4 is algebraically less than — I, and the velocity 
of rotation which corresponds to this case is given by the equation 

n** _ (Z-/Q (h-h) _ V+p)(p + h) 
h 2 ~ l+k p — l ' 

which gives 

2l(l+k) 
P — L ~ i +k > 

' w- 2naH ffl ( l + h w) 

w ~h(i-h)«l{p + h) l \ i-h'Vr 

780. When p is equal to — /, it is also equal to I s , that is, 

p = — l=I s . 

In this case 4 is algebraically greater than — /, and the veloc- 
ity of rotation which corresponds to this case is given by the 
equation 

n 2 a'_{l-l i ) (1,-1,) __(l-k) (k-k) 



h 2 ~ l+k l+l. 



J 



which gives 



2l(l+k) 



— 449 — 

781. If, in the preceding case, 4 is equal to the negative of 
I, it will also be equal to l 2 , that is, 

and, in this case, the elliptic integrals disappear from the equations, 
so that they become 



2 ~4 



n" a 
~h, 



= l — ll, 



4 = / 1 _(/ 1 -4-/)Tan 2 [A(^-T) v /(/+4)], 

and although the axis is constantly approaching the upper vertical, after 
passing the loiver limit, it never reaches the upper limit ; and if it begins 
at the upper limit it never recedes from it. 

782. In the simplest form of the problem of the spinning of the 
top, the extremity of the body is a point in the axis of revolution, ivhich 
is restricted to move, ivithout friction, in a horizontal plane. In this case, 
the equation (444 9 ) is still applicable, as well as (445 7 ), provided 
that the moments of inertia are referred to the centre of gravity 
of the top, and that I denotes the distance from the centre of grav- 
ity to the point in the horizontal plane. 

The equation (438 3 ) gives, in this case, with the notation of 
§772, 

(/ 2 -4 2 )/+r(l+^^-7, 2 4 2 )/=A 2 (^-4 2 )a-4); 

and if i// is eliminated by means of (445 7 ), 

P{lJ r lU!-I^ll)(=h\P-ll){l g -k)-n"a i {l z -l s f. 

The comparison of this equation with (445 19 ), shows that the 
limits of motion are the same as in the case of the gyroscope, and 

57 



— 450 — 

under the condition of the equality of l x and 1%, the extremity of 
the axis of the body describes a horizontal circle. The expressions 
of the time and of the azimuth of the axis are not, however, capable 
of expression by means of elliptic integrals, except in special cases, 
of which that of § 781 is one, and another corresponds to the case 
of 

■*-x 

783. "When the horizontal plane, to which the extremity of 
the top is restricted, is not smooth, the problem is usually more 
complicated, although when the friction brings the lower extremity 
to the case of rest, it reassumes the form of the gyroscope, and this 
is the modification of the problem which has been investigated by 
Poisson. In this case of the gyroscope, hoivever, the friction becomes an 
interesting feature of the problem, and has a peculiar effect upon the 
limits to ivhich the motion is subjected. Instead of the equation (444 9 ), 
the rotation about the axis of the body decreases uniformly, which is ex- 
pressed by the equation 

tf z = n — n 2 t. 

The area described about the vertical axis is also described in this 
case, at a uniformly decreasing rate, which gives instead of (445 7 ), 

(P — I 2 ) y' + — n' t) a 2 l s = n a 2 (7 3 — &). 

The power of the system is reduced by the friction about the 
body-axis, which is proportional to the angle %, and by the friction 
about the vertical axis, which is proportional to \\k If, then, the 
mean values of x\)' and £ for a small interval of time are denoted by 
w' m and £ OT , the equation of the preservation of power may be re- 
duced to 

(l*-l*)/+-f£=h 2 {l 2 -l 2 ) (4 + /o + ^)>. 



— 451 — 



in which 



2$r?ot= ji jf O' cos £) — n -£j§ y 

n' , , .. n a 2 l' s , , 

= p Wm * COS S m -j ¥ j 2 Xjl m t . 

The combination of this equation with (450 23 ) gives 

It is obvious from this equation, that if the friction about the 
body-axis vanishes, the height, to which the gyroscope ascends, 
diminishes at each oscillation. If, however, the friction about the 
vertical axis is destroyed, the height, to which the gyroscope 
ascends at each oscillation, increases when the body-axis is directed 
upwards in its mean position ; but this height diminishes when it 
corresponds to a position in which the centre of gravity is h^low 
the fixed extremity of the axis. In all intermediate positions, and 
when both the frictions remain, the increase or decrease of ascent 
depends upon the peculiar relations of the various constants. 

In the spinning of the top, the rounded point rolls upon the 
supporting plane, which induces an acceleration about the vertical 
axis which is the reverse of friction, and this is the principal cause 
of the ready rising of a top into the vertical position of apparent 
repose, known as the sleeping of the top. 



THE DEVIL ON TWO STICKS AND THE CHILD'S HOOP. 

784. Contrasted with the motion of the gyroscope is that 
of a solid of revolution of which, instead of a fixed point of the 
axis, the circumference of a section drawn through the centre of 



— 452 — 

gravity, perpendicular to the axis is restricted to move upon a 
point. A convenient type of this class of motion may be found 
in the familiar toy called the devil on tivo sticks. If the friction is 
neglected in this case, and the notation adopted from the preceding 
problem of the gyroscope, the rotation about the body-axis is found 
to be constant, and the equation for the preservation of area about 
the vertical axis is, by a slight reduction, 

Y sin 2 £ -\- n cos £ = B, 

in which B is an arbitrary constant. The principle of power gives, 
by reduction, the equation 

2 2 

i// sin 2 £ -|- £' = H-\- a sin £ , 

in which H is an arbitrary constant, and a is a constant which 
depends upon the form of the solid and the radius of the confined 
circumference. 

785. The combination of (452 9 ) and (452 M ) gives 

sin 2 £ £ /2 = (H-\- a sin £) sin 2 £ — (B — n cos £) 2 , 

from which it is obvious that, in the general case, sin £ cannot 
vanish, that is, the body-axis cannot become vertical. 

786. When B vanishes, and H is greater than a, we have 
the ordinary case of the devil on two sticks, and, in this case, there 
are three real values of sin £, for which the second member of 
(452 20 ) vanishes. Two of these values of sin £ are contained be- 
tween positive and negative unity, and one of them is positive, 
while the other is negative ; they give the limit of the motion of 
the axis, and correspond respectively to the cases in which the 
centre of gravity is below or above the point of dispersion, which 
latter is of course the actual case of the toy. In either case, the end 



r,Q 



— 45 

of the body-axis describes a curve ivhich is similar in form to the figure 8, 
and the apparent want of rotation about the vertical axis arises from the 
repeated change in the direction of rotation which occurs at each successive 
return of the bodg-axis to the horizontal position. 

787. When B vanishes, and II is greater than a, but satis- 
fies the inequality, 



H>%{hnaf 



n 



the three values of sin£, for which the second member of (452 20 ) 
vanishes, are all contained between positive and negative unity. 
The positive value is the upper limit of the inferior position of the 
centre of gravity, as in the preceding case, and as it would be if 
the inequalit}^ of this section were not satisfied, so that both the 
negative values were to become imaginary or equal. But the two 
negative values are the limits of motion, when the centre of grav- 
ity is higher than the point of suspension, and in this case the bodg- 

m 

axis describes a waving curve, and continues to rotate in one and the same 
direction about the vertical axis, ivithout ever becoming horizontal, ivhich 
phenomenon usually occurs in the devil on two sticks, at the beginning of 
the game, and before it has attained a sufficiently rapid rotation to assume 
a horizontal position. 

When H satisfies the equation 

R='6^nay — n 2 , 

the two negative limits of sin £ are equal, and correspond to a 
gyration of the body-axis about the vertical axis in a right cone. 
The motion which corresjDonds to a positive limit of sin £ in this 
case can be expressed by means of elliptic integrals. 
788. Whenever H satisfies the inequality 

H>B l + a, 



— 454 — 

the body-axis may become horizontal with the centre of gravity 
above the point of suspension, and in this position its gyration 
is positive or negative in conformity with the sign of B . If, more- 
over, n is greater than B , the vibration of the body-axis from the 
horizontal position extends so far as to reverse the direction of the 
gyration about the vertical axis ; but if n is less than B, the direc- 
tion of this gyration remains unchanged. 
When H satisfies the inequality 

H<B 2 -\-a, 

the body-axis cannot become horizontal with the centre of gravity 
above the point of suspension. 

789. The case of 

B = ±n, 

constitutes an exception to the conclusion of § 785, and it is 
obvious that in this case the body-axis may, and generally will, 
become vertical. 

790. The case of a hoop rolling upon a horizontal plane, is in- 
cluded in that of any rolling solid of revolution, but which is so 
formed that a circumference of the section of § 784 is restricted to 
roll upon a horizontal plane. The rolling condition is geomet- 
rically satisfied by the restriction that the point of contact with 
the plane is stationary during the instant of contact. If the nota- 
tion of the preceding sections is retained, and if / is the radius of the 
rolling circumference, the velocities of the centre of gravity in the 
directions of the body-axis and of a horizontal perpendicular to the 
body-axis are, respectively, 

It' andj£. 



— 455 — 

The equation of the power of the system multiplied by 2 I*, and 
divided by m becomes, then, 

i/sin 2 § + A r+ $1= H+ a sin § -f B n 2 , 
in which 

A=1 + PI 2 , 

a=z2gll%, 

ii is the initial velocity of rotation about the body axis, and II is 
arbitrary. 

The application of Lagrange's canonical forms to the preceding 
expression of the power gives the equations 

D t (Y sin 2 £ -f B v z cos j-) = 0. 

and by integration and reduction 

v z = n, 
y' sin 2 £ -]- i> « cos l—C, 

A sin 2 1 r = (5"+ a sin £) sin 2 t — (C—Bn cos £) 2 ; 

and it is obvious that these expressions coincide in form with these 
which were obtained in the investigation of the devil on tivo sticks, 
so that the various inferences made in that problem are applicable to the 
motion of the hoop. The analysis of the present problem is identical 
with that which was adopted by Nulty. 

791. When the hoop is gyrating with its plane in a position 
which is nearly horizontal, the cube and higher powers of sin £ may 
be neglected, in which case the equation (455 2 i) gives the integral 



2 j2 



r=s;+p* 



— 456 — 

in which it is sufficient to observe that £ is the initial value of £ 
and b is constant, so that the hoop constantly tends, by its inertia, to rise 
from this position, which, combined ivith the irregular action of friction, ac- 
counts for the peculiar forms of gyration, which frequently accompany 
the fall of the hoop. 



ROTARY PROGRESSION, NUTATION, AND VARIATION. 

792. The positions of the axis of rotation and of maximum 
rotation-area may be referred to a fixed axis, and the change of 
inclination to this fixed axis may be called nutation, while the gyra- 
tion about it is called progression ; and the change in the magnitude 
of the rotation, or of the maximum rotation-area may be called 
variation. ~ 

793. It is obvious from the simple principles of the computa- 
tion of rotation-areas, that an accelerative force which tends to give 
a rotation-area about an axis perpendicular to the axis of maximum 
rotation-area, does not cause a variation of the rotation-area, but 
only a motion of the axis so as to incline it in the direction of the 
accelerative axis. Hence if the accelerative axis is perpendicular to the 
fixed axis as well as to the axis of maximum rotation-area, progression 
is produced ; if it is in the common plane of the fixed axis and axis of 
maximum rotation-area, while it is perpendicular to the latter axis, nutation 
is produced ; if it is in the direction of the axis of maximum rotation- 
area, variation is produced. 

The three directions of the accelerative axis, which correspond 
to the respective production of progression, nutation, and variation 
are mutually rectangular ; so that it is easy to determine the rela- 
tive tendency of a given force to these different modes of action. 
This neat analysis is derived from Poinsot. 



— 457 — 

794. If the accelerative axis is constantly perpendicular to 
the fixed axis, and also to the axis of maximum rotation-area, the 
motion will be wholly that of progression, of which mode of action 
a fixed type is presented in the precession of the equinoxes, the 
discussion of which problem must be reserved for the Celestial 
Mechanics. If the accelerative axis is constantly in the plane of 
the fixed axis and of the axis of maximum rotation-area, while it is 
perpendicular to the latter axis, the motion is exclusively that of 
nutation, and this form of action is well exhibited in the friction at 
the point of the top as it rolls upon the horizontal plane. 



ROLLING AND SLIDING MOTION. 

795. A special example of the case of rolling motion has 
been considered in the hoop, and the mode of analysis tvhich ivas there 
adopted can be applied to the general investigation, as it has been done 
by Nulty. Thus, let the axes of x, g, z have the same directions 
with the principal axes of the rolling solid, let x g ,g s , and s g denote 
the coordinates of the centre of gravity of the solid, and x T , g T , and 
z T those of its point of contact with the surface upon which it rolls. 
The condition of rolling without sliding gives the equation 



*£ = C^-^K-- (*,-*,) 4 



yi 



with the similar equations for the other axes. The expression of 
the power is 

T= h m Z x [i [1 + (g-g g f + (s-s s ) 2 ]- 2 (y-g s ) (*—*,) V y £)], 

from which the equations of nutation can be readily obtained by 
Lagrange's canonical forms. 

796. If the solid slides upon the surface, it' still remains in 

58 



— 458 — 

contact with the surface, so that the point of contact does not 
move in the direction of the normal to the surface. If the direc- 
tion of the normal is denoted by N, this condition is expressed by the 
equation 

which is given by Anderson. This is the only condition, to which the 
motion is subject, in the case of perfect sliding motion. 

797. When the sliding: is accompanied ivith friction, the friction 
may be regarded as a force proportional to the pressure applied at the point 
of the solid, ivhich is in contact tvith the surface, in a direction opposite to 
that of its motion. 

When the velocity of rasure is destroyed by friction, the motion 
ceases to be sliding and becomes a rolling motion, in ivhich form it 
continues as long as the force of friction exceeds the accelerative force in 
the direction of friction. 



CHAPTER XIII. 

MOTION OF SYSTEMS. 

798. The motion of every system is necessarily subject to the 
Law of Poiver, expressed in § 58, to the law of the motion of the centre 
of gravity of § 452, and to the laiv of areas of § 753. These three 
principles not only apply to the whole system, but to each portion 
of it considered as a system in itself. 

799. The various forces which act upon a system are often 
quite different in the magnitude of their effects, so that they may 



— 459 — 

be considered from this point of view as different orders of force. 
In a first investigation, all but the forces of the first order may be 
neglected ; and in subsequent approximations the forces of the 
inferior orders may be successively introduced, as disturbing forces 
and their various effects may be determined as perturbations of cor- 
responding orders. 

800. The separation of the system into partial systems is 
closely connected with this subdivision of the forces, for it may 
easily be seen that the forces, which are of chief importance in 
the whole system, or some portion of it, are least active in other 
portions of this system. Whenever, for instance, the parts of any 
portion are so isolated from the rest of the system, that their 
relative changes of position are of small influence out of the 
portion, they should be treated by themselves as a partial system, 
and, relatively to all the other parts, may be considered as con- 
densed upon their common centre of gravity. 



LAGRANGE S METHOD OF PERTURBATIONS. 

801. The method of perturbations which originated with La- 
grange, and which depends upon the variation of arbitrary constants, 
deserves the first consideration from its surpassing elegance ; and 
it is the natural introduction to the other modes of investigation. 

Suppose, then, that a complete system of integral equations 
is obtained, when all the forces but those of the first order are 
neglected, and let one of these equations involving a single arbi- 
trary constant be denoted by (199 20 ). Let 

£2 denote the potential of the forces of the first order, and 
W that of the forces of the inferior orders, 



— 460 — 
and the equations of motion (166a_ 3 ) assume the forms 

If the constant member of (199 20 ) is now assumed to vary, its 
derivative is 

D t a i = Z v {DJ i D n T) = Z k iZ v {DJ i D 11 f k )D f W-\, 

for by (199 25 ) 

= S n (B r/ f t D u II- DJ { D ri II) . 

The condition that W does not involve co gives algebraically, 

' z k (i) u f k n f y) = o., 

and the notation 

Af = DJ k DJt — D n fi DJ k , 

Bf = Z v Af, 

gives in combination with (460 8 ) 

in which a may be substituted for its equal / in the second member. 
802. The integrals of (460^) obtained with the omission of 
the forces of the inferior orders, admit of arbitrary variation of the 
arbitrary constants, so that if such variations taken with reference 
to arbitrary elements which may be denoted by x and 1, the cor- 
responding variations of (460 2 _^) with the omission of the terms de- 
pendent upon W are 

D t D K a> = -I) v I) K II, 



— 461 — 
and similar equations for X which give 

= Z v {D 1 D (J HD K to+D x D v HD Kn -D K DJID^-D K D v HD^) 
= D,D K R-I) K I),R=0, 

so that if 

C^ 1 does not involve the time explicitly. 

803. If x is the element of actual variation of the arbitrary 
constants when the inferior forces are introduced, which element 
may be expressed as the time when it is so connected with the arbi- 
trary constants, as not to cause ambiguity, the variations of the 
equations ^GO^), give 

so that D x W does not involve the time explicitly. When x and I 
are changed to a t and a k , it is sufficient to retain i and h in the 
notation C£' ] , so that it is apparent from (461 18 ) that 

By elimination from the equations represented in the preced- 
ing form, the value of D t a { can be obtained identical with that of 
(460 2 o), so that it is evident that B$ does not contain the time ex- 
plicitly. It is also apparent that 

except when 



— 462 — 
in which case 

804. The independence of B { 1 ] of the time in an explicit form, 
renders it possible to compute its value for any instant, and the 
value thus obtained is universally true. Thus in the especial case 
in which the arbitrary constants are the initial values of rj, w, etc., 
the values of B [ j^, computed for the initial instant, are easily seen 
to vanish when the k and i refer to different points of the system ; 
but when Jc and i refer to the i] and tu of the same point, the value 
of B [ k ] is positive or negative unity, so that 

D t co = D v T. 

In the case of rectangular coordinates these equations become, 
for either axis, 

D t x' = D x W. 

805. The especial variation of the constant H may be de- 
rived from the equation (171 7 ) which gives 

n t ir=z v ii) v TD tn ) = i) tTi T, 

provided that t is intended to express the t which is involved in 
any of the quantities denoted by rj. This development of the 
variation of the arbitrary constants is taken from Lagrange. 



LAPLACE S METHOD OF PERTURBATION. 

806. The values of w, ij, etc., can be substituted from the 
first integrals directly in the first form of the second member of 



— 463 — 

(460-), and the integral values of c^ which are then obtained can 
be introduced into to, 17, etc., as a second approximation to their 
values. This mode of analysis is especially useful when the equa- 
tions of the first form are linear with reference to rj, o), and their 
derivatives. For in this case it is apparent that the functions de- 
noted by fi are linear with reference to rj, to, which may be demon- 
strated in the following manner. Let rj i} to L , etc., be special values 
of ?], to, of which there must be as many independent values as 
there are equations expressed by (460 2 _4). The arbitrary constants 
a { may then be such that the complete values of rj and co are 

to = S 4 (a t tot) ; 

whence the values of a L assume, by elimination, the linear form in 
reference to 1], to, etc. The values of D a f i} are then functions 
of t, and do not involve rj, to, etc. If D *P represent forces, which 
are also functions of the time, the integrals of (460 7 ) can be com- 
pletely obtained. By the substitution of these values of a t thus 
obtained in the expressions of rj, the complete values of rj are ob- 
tained, which often admit of useful modification, and the success of 
the method depends upon the skill with which this modification is 
effected. 

807. A special case of frequent occurrence in the problems 
of celestial mechanics is one in which 

10=1]', 

H= corf -j- i a 2 if. 
The value of the integral in this case is, for a first approximation, 

rj == a cos at -\-a x sin a t , 



whence 



— 464 — 

o) = — a a sin a t -j- a a x cos a t. 
a = rj cos at sin a t =f, 

a x = rj sin a t -j- - cos at=f 1 . 



The values of the constants obtained by integration of (4602_^) ; 
are increased to 



and 



<*-\j;{D v T^at).. 



<*i + -/(.Oleosa*; 



so that the complete value of r\ is 

i'j=za cos at -f- «isin at — j (D v Tsmat) _j_ sma / (j)^ Wcosat). 

808. The disturbed motion of the ordinary projectile ex- 
hibits an easy example of change of form. In this case, by the 
introduction of rectangular coordinates in which the axis of x is 
horizontal, and that of y vertical, the equations are 

D t x' = D x W, 

D t y' = -g + D y T, 
whence 

% = at + a 1 + tf t D x W— f t {tD x W) 

= at + a x -\-f?D x T, 
y = —:hgf+azt + a z + tf t D y V—f t (tD y W) 

= — i< / ? + a 2 t-\-a 3 +f?I) v ¥. 



— 465 — 



Hansen's method of perturbations. 



809. If V t denotes any function of the time and of the arbi- 
trary constants in the undisturbed orbit, its value in the disturbed 
orbit may be obtained, from the integration of the equation 

by the substitution of t for % after the integration is performed. 
In the performance of the integration, the arbitrary constants are 
to be regarded as variable, and the value of V t in the undisturbed 
orbit is to be taken for the initial value of V T . This introduction 
of x for t constitutes the first principle of Hansen's method of pertur- 
bations. 

810. The application of this method to the example of § 808, 
gives, for the values of x and y 



x 



: . = at + W+f£(*--t)I) m ¥], 



t 
o t 

811. In the example of § 807, the value of rj given by this 

method is 

t 

■}] = a cos a t -\- «! sin a t J sin a (t — t) D v T T L 



in which the form of notation is slightly modified so that no subse- 
quent change of r to t is necessary. A case, which often occurs in 
connection with this example, is worthy of notice ; it is when 

D n ¥=hcos (mtf-fe), 
59 



— 466 — 

in which case the value of rj is 

rj = a cos a t -J- a x sin a t -J- - 2 2 cos (m ^ -|-e) . 



In the special case of 

m = a, 
this value of ?j becomes 

7] = a cos a t -\- «! sin a t -f- — tf A sin {a t -J- e) . 

812. If the function F increases with the time from negative 
to positive infinity, so that for all values of t 

D t V>0, 

there is an instant which may be denoted by z, for which the un- 
disturbed value of V coincides with its disturbed value for the in- 
stant denoted by t. The corresponding value of z T is a function of 
both t and t, which may be introduced into V T instead of r, but 
after this substitution all the changes in the value of V T must arise 
from those of z T , so that 

D t V=D V D t z r , 
D T V T = D V D r z T , 

T T 

and the differential equation for the determination of z T is 

In the integration of this equation, % must be taken as the initial 
value of z Ti whence, for the first approximation, 

D T z T =l. 



— 467 — 

After the integration is performed, the value of z is derived from 
that of z T by changing t to t . 

813. The disturbed value of any other function, U may be 
partially obtained by the substitution of z for t, and, since 

D r u T =D T u Zr + D. r u Zt D t z t , 

the residual portion is obtained from the equation 
D t U z =D t U t —D Zt U z D t z T , 

D T U T x -. D U z 






'T? 



by changing % for t after the integration is performed, and complet- 
ing the integration, so that JJ Z may be the value of U z when t 
vanishes. 

This introduction of the disturbed time, which 'is denoted by z, con- 
stitutes the second principle of Hansen's method of perturbations, and 
upon the skilful use of the two principles thus developed, com- 
bined with an appropriate choice of coordinates, depends the suc- 
cess of this highly ingenious and original method. 

814. It is obvious that, in the first approximation, 



d t u Zt =o, 



so that the last term of (467 n ) disappears for this approximation. 

815. If V is such a function that it can be expressed in terms 
of r], etc., without involving a>, etc., or t, it follows from §801, 
that the second member of (46 5 8 ) vanishes, when % is changed to 
t, so that this must also be the case with the second member of the 
equation derived from (466 27 ), 



D t z T 



D T *r 



- ( Da i Vr n \ 



— 468 — 

The value of the first member of this equation can therefore be 
obtained by the integration of the equation 

ID z \ i D r VrD ai D T V T -D ai V T D\ V T , 



T~T 



provided that the integration is completed in conformity with the 
previous condition. 

816. If one of the arbitrary constants, which may be denoted 
by § is so involved in V that 

in which K does not involve the time, or if the form of V is 

the corresponding term of the second member of (468 4 ) is 

D 2 V 

The corresponding term of the second member of (467 n ), if U has 
the same form with V — /3 in (468 u ), is 

DjUr D t (5. 



D T V T 



817. If one of the arbitrary constants, which may be de- 
noted by y is so involved in U that U — y may be expressed as a 
function of V without explicitly involving y or t, the corresponding 
term of (467n) is reduced to 

Ay- 

818. The further development of the methods of perturba- 
tions depends upon the peculiarities of the problem to which they 



— 469 — 

are applied. But the example, to which they are most appropriate, 
is that from which they have derived their origin, the motions of 
the bodies of the solar system, so that their ampler discussion is re- 
served for the Celestial Mechanics. 



SMALL OSCILLATIONS. 

819. When the motion of a system is restricted to small 
oscillations about a position of equilibrium, the quantities t], etc., 
may be supposed to be so small that the terms of T and £2, which 
are of more than two dimensions in "reference to these quantities 
and their derivatives, may be neglected. 

The value of T may, then, by (165 8 ), be expressed in the form 

in which the quantities denoted by T^ i] , are constant. 

If the values of r\, etc., are supposed to vanish for the position 
of equilibrium, the derivative of £2 with reference to either of 
these variables vanishes for the same position, so that £2 must have 
the form 

£2 = £2 Q -\-2 h)i (£2fr ]k r }i ), 

in which the quantities, denoted by £2$, are constant. 

The equations of motion, derived from Lagrange's canonical 
forms, are, therefore, represented by 

that is, ilwj constitute a system of linear differential equations ivith constant 
coefficients. 

820. It follows from the linear forms of these equations, that 



— 470 — 

the various systems of values by which they are satisfied, can be 
combined by addition into a new system. This is the mathematical 
expression of the important physical law of the possibility of the 
superposition of small oscillations. 

821. With the notation 

af = T™D 2 — £lf, 
the equation (469 28 ) assumes the form 

If, then, there are m of the quantities r\ , etc., if — n 2 is one of the 
values of D 2 which satisfies the equation, expressed in the notation 
of determinants, 

any system of values of ^ is expressed by the equation 

i] i = E i sm(nt-{-e n ), 

in which e n is 'an arbitrary constant, and the constants E { have a 
common arbitrary factor. The mutual ratios of the quantities E i 
are determined from the equations derived from (470 10 ) by the 
substitution of — n 2 for D 2 , and E for rj. Hence, by § 340, E t is 
determined in the form 

E i = E n D^%> m , 

in which E n is an arbitrary constant. 

822. By the combination of all the values of n, the complete 
value of rj t is 

Vl = Z n \E n Df ^ m sin (n t + «.)] ; 

but it is evident that only those values of n should be retained 
for which the values n 2 given by (470 14 ) are real, positive, and 



— 471 — 

unequal. For all other values of n 2 , the time t would be intro- 
duced into the value of i] t in such a way that it would indefi- 
nitely increase. .It is plain, therefore, that the only values of n, 
which can be- retained in (470 2S ), are those which correspond to 
elements of stability, so that if the elements i] are selected with 
a due regard to the conditions of equilibrium, those which corre- 
spond to the unstable equilibrium will disappear of themselves 
with the rejection of the corresponding values of n. 

When the position of equilibrium is stable ivith reference to all of its 
elements, all the m values of n 2 are real, positive, and unequal. 

823. The forms of T and S2 of § 819, lead, by inspection, to 
the equations 

T U] = T- lk] 

and the equation (469 28 ) gives, for each value of n, 

if n written as an accent indicates a special value of n, to which the 
functional form is applicable. If £„ is determined by the notation 

and if the equations, represented by (469 28 ), are added together 
after being multiplied by El n \ the sum is 

If, moreover, T n denotes the value of T when ^' is changed to Ej-" ] , 
the value of £„, given by integration, is 

i n = T n sm(nt-\-t n ). 
The elements £ thus obtained, correspond to the independent ele- 



— 472 — 

ments of stability which affect the position of equilibrium, and 
embody the true analysis of the various forms of oscillation of 
which the system is susceptible. When the different values of n 
have a common divisor, the oscillation is evidently periodic. 

This investigation of the theory of small oscillation coincides, 
in substance, with that of Lagrange. 

824. The importance and variety of the forms, in which 
oscillation and vibration are physically exhibited, give peculiar 
interest to the mechanical discussion of this subject. But the mode 
of analysis is so dependent upon the form of the phenomena, that 
the special researches are reserved for the chapters to which they 
are appropriate. 



A SYSTEM MOVING IN A RESISTING MEDIUM. 

825. When a system moves in a resisting medium, the law 
of resistance may be regarded as dependent upon the velocity, so 
as to be the same for all the bodies, but it may vary by a constant 
factor from one body to the other. If this constant factor for the 
mass mi is denoted by k i} and if T^ is the function of the velocity 
Vi, the resistance to the mass nti moving with the velocity v t is k t V t . 
If, then, rectangular coordinates are adopted, the equations of 
motion assume the form 

t l nti x i l % v { 

The corresponding form of the equation for the determination of 
the Jacobian multiplier is, by §§ 402 and 451, 



D t log Mifk = 2< [h Z x D X{ M\ 



— 473 — 
This equation becomes, when the motion is unrestricted in space, 

D t log ,** = S t [k t (2 J + A, v)] ; 
when the motion is in a plane, 

D t logM = S i [k i (^-\-D Vt v)] ; 

when the motion is in a straight line, 

D t \og^ = Z i {lc i D Vi V i ). 

826. It is evident from the linear form of these equations, 
that the multiplier can he separated into factors, each of zvhich shall inde- 
pendently correspond to a term of V } . 

827. When the resistance is constant, and the motion in a 
straight line ; or when the resistance is inversely proportional to 
the velocity, and the motion is in a plane ; or when the resistance 
is inversely proportional to the square of the velocity, and the 
motion is unrestricted in space, the multiplier becomes unity. In 
either case of motion, a term of the corresponding form may be 
added to the resistance without affecting the multiplier. 

828. When tJie resistance is proportional to the velocity, the value of 
the multiplier in the case of unrestricted motion is 

in the case of motion in a plane it is 

asito = c 2tl i k i ; 
and in the case of the straight line it is 

njbMd = C f S « k i . 

60 



— 474 — 

All these results, with regard to the multiplier, are derived 
from Jacobi. 

829. When the resistance is proportional to the square of the velocity, 
the value of the multiplier for motion, which is unrestricted in 
space, is 

cdk ■==(•*?« (*««<).; 

for motion in a plane, it is 

and for motion in a straight line, it is 

<&MD = c 2 ' z i( k i s i>. 

830. The sum of the equations (472 26 ), multiplied by m { x\, is 

D t (F—£2) = — S t {h m t V, »<) . 
When Vi has the form 

1 Vi' 

the integral of this equation is 

T — 12 = — S { [h nii ( Si + a t t)] . 

831. When there are no external forces acting upon the sys- 
tem, the sum of the equations (472 26 ) for each axis multiplied by m i} 
if x g refers to the centre of gravity, is 



\ Mi D\ x g = — S t (wii ki V~J . 



If the resistance is proportional to the velocity, the integral of this 
equation is 

S { nii {D t x g — A) — — Si (nii Jc t x t ) , 



— 475 — 

in which A is an arbitrary constant. If k t has the same value for all 
the bodies, the complete integral is 



k x„ — A = B c 



-kt 



in which B is an arbitrary constant. 

832. The introduction of polar coordinates, and the substitu- 
tion of A [ J ] for the product of the area described by m f about the axis 
of &, multiplied by the mass m i} give for the corresponding equa- 
tions of motion 

& t Ay = D m a — kfiD t Af: 
When there are no external forces, the sum of these equations is 

When the resistance is proportional to the velocity, the integral of 
this equation is 

D t S i Af=C—2 i {k i Af), 

in which C is an arbitrary constant, which vanishes if the area van- 
ishes with the time. If k { has the same value for all the bodies 
the next integral is 

2 i Af = B(l—c- kt ). 

So that the rotation-area instead of being proportional to the time is pro- 
portional to 

l — c- kt , 

hit the position of the axis of maximum rotation-area is not affected by 
this uniform mode of resistance, which proposition is from Jacobi. 



— 476 



THE CONCLUSION. 



833. In the beginning, the creating spirit embodied, in the 
material universe, those laws and forms of motion, which were best 
adapted to the instruction and development of the created intellect. 
The relations of the physical world to man as developed in space 
and time, as ordered in proximate simplicity and remote complica- 
tion, in the immediate supply of bodily wants by the mechanic arts, 
and the infinite promise of spiritual enjoyment by the contempla- 
tion and study of unlimited change and variety of phenomena, 
are admirably adapted to stimulate and encourage the action and 
growth of the mind. True philosophy begins with the actual, but 
may not remain there ; it yields sympathetically to the projectile 
force of nature, and earnestly forces its path into the possible, and 
even into speculations upon the impossible. But whenever the 
initial impetus is exhausted, the philosopher may not be content 
to remain stationary, or merely to turn upon his axis. He, then, 
descends to the world of sensible phenomena for new instruction 
and a stronger impulse. Let such be our method. In the present 
volume the attempt has been made to concentrate the more im- 
portant and abtruser speculations of analytic mechanics clothed in 
the most recent forms of analysis, and to make a few additions, 
which may not be rejected as unworthy of their position. Much, 
undoubtedly, remains imperfect and unfinished, for it cannot be 
otherwise in a science which is susceptible of infinite improve- 
ment ; and much must soon become antiquated and obsolete as 
the science advances, and especially when we shall have received 
the full benefit of the remarkable machinery of Hamilton's Quater- 
nions. But it is time to return to nature, and learn from her actual 
solutions the recondite analysis of the more obscure problems of 



— 477 — 

celestial and physical mechanics. In these researches there is one 
lesson, which cannot escape the profound observer. Every portion 
of the material universe is pervaded by the same laws of mechanical 
action, which are incorporated into the very constitution of the hu- 
man mind. The solution of the problem of this universal presence 
of such a spiritual element is obvious and necessary. There is one 
God, and Science is the knowledge of Him. 



APPENDIX. 



NOTE A. 



ON THE FORCE OF MOVING BODIES. 



It is remarkable, that, notwithstanding the convincing argu- 
ments of Leibnitz, the force of moving bodies is almost universally 
introduced into systems of analytic mechanics as being proportional to 
the velocity, instead of to the square of the velocity. Some philos- 
ophers, in quite an unphilosophic spirit, have stigmatized the early 
discussions of this subject as a war of words, as if the eminent 
geometers who entered into it could have been so deficient in their 
powers of logic and analysis. The great objection to the propor- 
tionality of the force to the velocity is derived from the necessity 
which it involves of regarding force in one direction as beino; the 
negative of that which is in the opposite direction. On this ac- 
count, when a body or system rotates without any motion of transla- 
tion, its aggregate force vanishes, so that such a motion would seem 
capable of being produced without any expenditure of force, and 
this statement has actually been made in some works upon astrono- 
my. Leibnitz proposed as test propositions the transfer of motion 
from body to body in various forms, in all of which he supposed the 
whole force to be transferred from one body to another of a dif- 
ferent weight without any external action. But it is evident from 
the law of preservation of momentum that such a transfer is im- 
possible, and, therefore, this test cannot be practically applied. If, 
however, in the case of the impact of an elastic body upon a 



— 480 — 

heavier one at rest, the striking body is held fast, as soon as it comes 
to instantaneous rest by the transfer of all its motion to the other 
body, the subsequent action of the elasticity must finally cause the 
body which is struck to move forward with a velocity inversely 
proportional to the square root of its mass. The external effort 
applied to the system in this case to hold the body at rest, arises 
from the force with which the elastic spring of the bodies is com- 
pressed, and is therefore an evidence of such a compression, and 
a proof that there has been an expenditure of force in its produc- 
tion, although the momentum of the system is not changed until 
the body is held. If, again, a spherical ball were to be impelled into 
a cylindrical tube of the same diameter, which terminates in an- 
other cylinder of a different diameter, but which containing a ball 
that exactly fits it, and if the included air acts as a compressed 
spring, it is easy to imagine such a mutual proportion of the parts 
and weights that the second ball shall leave the cylinder at the 
very instant when the first ball arrives at a state of rest, and when 
the air has returned to its initial density. In this case the whole 
living force of the first ball passes without increase or diminution 
into the second ball, and the momentum is not preserved. It is 
true that an external force is required to keep the cylinders in 
place, but this is a mere pressure, which is no more entitled to be 
regarded as an active force than is the centrifugal force, or any of 
the modifying forces which are represented analytically by equa- 
tions of condition. Seeing, then, that by admitting the square of 
the velocity to be the true measure of the force of a moving body, 
the fiction of negative force is wholly avoided, and the funda- 
mental principles of mechanical problems are reduced to their 
utmost simplicity, there seems to be sufficient reason to reverse the 
modern decisions, and return to the higher philosophy of Leibnitz. 



— 481 



NOTE B. 

ON THE THEORY OF ORTHOGRAPHIC PROJECTIONS. 

For the convenience of students, the theory of orthographic 
projections is here condensed into a few simple formulae. 
The projection of a line a upon another line b is 

# 6 = a cos l . 

If many successive lines represented by a t , are so united that 
each line begins where the previous line ended, and if the last line 
terminates where the first began, the sum of the projections is 

Si (^ cos *.) = 0. 

If there are four of these lines, and if the three first are mutually 
rectangular and parallel to the axes of x, y, and z, this equation 
becomes 

S x {a x cos I ) -f- tf 4 cos * = . 

But it is evident that a x is the projection of — « 4 upon the axis 
of x, whence 

CL X = fl5 4 COS „ , 

and if the subjacent 4 is now omitted as unnecessary, this equation 
gives 

cos* = S x (cos" cos*), 

of which the equation 

l = ^cos 2 «, 
is a particular case. 

These equations may be applied to the projections of plane 
areas, if each area is represented in a linear form by the length of 
a line which is drawn perpendicular to it. 

61 



ERRATA. 



Page 


For 






Read 


12 9 


axes 






axis. 


lo^o and 15^ 


The signs of the second members should be reversed. 


15 24 


acute 






right. 


26 8 


I 






l v 


30 17 


these 






those. 


40 18 


resultant 






resultant moment. 


40,3 


different lines 






opposite directions. 


4122 


force 






resultant of the forces. 


42., 7 


0' with reference to 







O with reference to 0'. 


Wi 


X, 






x x . 


51 4 


y 






v. 


55 14 


POINT UPON A DISTANT MASS 


MASS 


UPON A DISTANT POINT. 


57 13 


4 






.*• 


5722 an d 5728 


\ 






I- 


*592 


cos-^ 






COS jy. 


59 21 and 60^ 


four 






eight. 


5933 an d 60 21 


two 






four. 


73 16 


surface 






surfaces. 


83 2 


\ 






K- 


85 6 


D% 






B r 


85 21 


i 






4. 


86 u 


4rt 






AnK 


8622 


\ and \ 






1 and 2. 


88 10 


-A) 






+ ^)- 


90 18 and 9O20 


Jfm-l 






H n ~\ 


90^ 


(cos (m — 1) 






cos (to — 1) »/. 


9125 


See note on page 


356. 




98 24 


89 7 






89^. 


J J 10 


r n 






»■». 


100; 


independent of 






dependent upon. 


101^ twice 


+ r 






+ 2r. 



* This correction only applies to some copies. 



484 


ERRATA. 




Page 


For 






Read 


l^M, 9, 16, 18 


3 






6. 


107 n 


Al 






A x ,. 


10729 


(104 19 ) to the form 


(107 l6 ) 


(104 9 ) to the form (1070 


Ulio 


Sn 






^m * 


111 13 


— 2 






2. 


Hll3 


( r \-{m-l) 






( r \— (m.— 2)^ 


111 


k 






h 


-*- *■ ■*■ 17, 20 


Fmr m ~ l 


fmr™— 2 • 


Hll9 


72 






24. 


H7» 


42 12 






42 2 2- 


H7 22 


4Px 






Au x . 


H7 16 


for another point of the 
is near the former 


body 
point 


which 


arising from this motion. 


H7 3 i 


dele Ax-=. 








120 7 


\p 






\n. 


120 20 


similar to 






like. 


121 x 


119, 






11 8 3 . 


122^ 


n 






\n. 


123 s 


above 






about. 


126 u 


put 






but. 


127 8 


in order 






in order that. 


127 M 


place 






plane. 


139 16 


tan 






cot. 



14023 r C" 

140^ C» 

141 16 _ 20 sec 2 ^ 

148 ls cot £ 

148 22 takeF'K 

149 3 + sin 2 tj 

1 49 6 cos 2 ra == cos 2 a 

149 7 sec 2 01 
1 49 ]8 cos 2 6 sin 2 tj sin 2 cp cos 2 6 -j- sin' 2 tj sin 2 8 cos 2 d — sin 2 rj sin 2 cp cos 2 6 -J- sin 2 j? sin 2 5 cos 2 qp. 



BcB. 

y 
cB. 

sec£. 

tans. 

take F' L. 

COS 2 Tj. 

cot 2 01 = cot 2 « . 
cos 2 01 . 



149 19 

15028 

150 30 twice 

150 31 

151! 

151 3 

151, 

151, 

151 OT 



sin 2 tj sin cp 



sin 2 tj sin 2 cp. 



0' + 

-j-tan/3 
0' — 

0' + 

— cos 2 j3i cp' 

cot <jp' 



— tan/?. 

0' + . 
0'— . 

— cos 2 j3 , qo'. 

(cotg>' + ±<p'). 









ERRATA. 


48 


Page 




tor 




Bead 


152 24 




a'—yo 




V cos 2 ^ 


152^, 




— 




+■ 


152 31 




cp"- 




9>"+- 


156,, 




prolate 




oblate. 


157! 




oblate 




prolate. 


173* 




for the 




for two. 


173^ 




determinate 




determinant. 


175 3 




i^>n — 1 




t>re — 2. 


175 a 




a)> k 




<4>* 


178 i3 




-B(m) 




.bm. 


180 4 




®n 




«„ 


180 7 




»: 




*:• 


19028 




B 




SB. 


191 2 




f 1 






191 


The sections 366 to 369 should be limited 


m<^n — 1. 




by the condition that 


191* 




080B{.... 




<@ <f J <@^ 


191* 




(-r^: +1 


( )<»+i> <»+i> g$£_ 1# 


192,0 




(_)» + ! 




/" \mre 


192 13 




■in — 1 




»l t. 


192, 3 




( \»+i 




/ \JM» + (i+l)(m+» + l) 

199a. 


200 1C 




199 8 




202 10 




2 




n 

2. 


203 g 




340 




339. 


215 19 




D Xi 




Z>^. 


^ 1 9l9, 24, 


22O3 


X 1 , X 2 . . . , 


..a; 


^"1 > 3J 2 • • • • "n • 


222 18 




200 2 




216a. 


22^ 




formal 




normal. 


226 u 




1—1 




1. 


227 2 




(210 3l ), the 




(210 31 ). The. 


227 9 




F 




^- 


228ao 




D x 




D Xl . 


228 21 




x i 




D*. 


231 25 




187 10 




189,. 


233 3 




21 6 U 




231 14 . 


234 5 , 7 




D Xk ksM* 




A* onA (i >. 


238,9 




2 k 




■Z». 


239 




vJD^ 




jr. 



486 



ERRATA. 



Page 

239 9 
247, 
247 K 
250 25 

258^ 

26^ 

262 IS 

262^ 

263 21 

265 9 

277^ 

279^ 

281 a 

281™ 



14, i» 
20 



282 2 

282; 

285 : 

287 ]0i 16 

287 12 

289 19 

317 6 

328-354 

328 9 

330 4 

348 22 

352 n 

364 19 

369-370 

377 30 

o93 2 3 

426,, 

431 19 

472 31 



Q 

uniform 

h 

P. 
p' 



sin cp 

Cot 
ht 
Ra 

±T 

correct 
+ « 

2 

i 

h 

b-\-2b 

Brachystochrone 

is confined 

329 9 

589 

sin v 

359^ 

Tachystotrope 

level 

sin [ -« 

<p\ 

.Q 2 



Read 

uniform and constant in direction. 

k. 
b 

T 

9°. 

2 a 2 . 
,+. 
cot. 
cos cp. 

v/[A(^-«)]- 

Tan. 

(Jc t — 2 a) . 

Ra ffl • o 

± -V sin 2 a. 

9 * 

nearly correct. 
— cc. 

^. 

5 + 2/3. 

jr. 

Brachistochrone. 

is not confined. 

329 3 . 

588. 

cos V . 

359 3 . 

Tachistotrope. 

given. 

sin [ - 1: . 

cp'. 

fio- 



ALPHABETICAL INDEX. 



Page 
Abel, method of investigating the holo- 

chrone, ..... 356 
Acceleration of rotating cylinder upon 
which a body moves, when it is 

uniform, 254 

of a point by a moving line, . 247 
Action and Reaction, . . . .132 
of moving bodies, . . . 162 
principle of least, . . . .16 7 
Anderson on rolling and sliding motion, 458 
Archimedes, spiral of, described by ac- 
tion of central force, . . . 384 
Area, constant area described by a 
point upon a surface of revolu- 
tion 412 

in the motion of a free point, . 424 
constant, when all the forces are 

directed towards the axis, . . 425 
of rotation, .... 433 
conservations of, . . . . 434 
of rotation for a principal axis, 436 
of rotation when it is a maximum 

for a solid, . . . . .437 
of rotation expressed by Euler's 
equations for the motion of a 

solid, 437 

of rotation, its axis when it is a 
maximum for the free motion of 

a solid, 439 

of rotation described by the gyro- 
scope about the vertical axis, . 445 
of rotation of gyroscope affected 

by friction, 450 

of rotation of the devil, . . 452 
of rotation of the devil, when it 
vanishes, ..... 452 





Page 


Areas, principle of, in a moving system, 


. 458 


Astronomical Journal, see Gould. 




Asymptote of the brachistochrone of ir. 


i- 


finite branches, . 


. 333 


Attraction of an infinite lamina, . 


46 


of an infinite cylinder, . 


. 49 


of a point upon a distant mass, 


. 55 


of a spherical shell, . 


56 


of a Chaslesian shell, . 


. 58 


of an ellipsoid, .... 


69 


of a spheroid, 


. 88 


Axis of rotation, ..... 


12 


of rotation, instantaneous, 


. 19 


of gravity, .... 


50 


of principal expansion, 


. 118 


of inertia principal, . 


436 


instantaneous in a body and in 


space, .... 


. 439 



B. 

Bailey on the force of resistance to the 

motion of the pendulum, . . 291 
experiments on the motion of pen- 
dulum of various forms in air, . 311 
Barnard on the gyroscope, . . 447 
Bary trope discussed, . . . .370 
Bernoulli, John, on the synchrone, . 373 
Bernoulli, James's, lemniscate, . . 380 
Bertrand on the tautochrone, . . 364 
Bessell on the resistance of the pendu- 
lum, 292 

experiments upon the seconds' 

pendulum, ..... 298 

Bobillier catenary on cone, . . 153 

catenary on sphere, . . .157 

Bonds of union of a rigid system, . . 126 

Bonnet, theorem of combination of forces, 430 



488 



ALPHABETICAL INDEX. 



Booth, elliptic integrals, . . .147 
Boeda, experiments on the seconds' pen- 
dulum, 296 

Brachistoehrone, 328 

on the surface of revolution, . 334 

Brass sphere vibrated by Bessell, . 298 
spheres, cylinders, and bars vi- 
brated by Bailey, . . .311 

C. 

Canonical forms of the equations of mo- 
tion, 163 

Catenary, 134 

on surface of revolution, . . 143 
on right cone, .... 144 
on ellipsoid, . . . . .154 
on equilateral asymptotic hyper- 

boloid, 159 

curious relation to the motion on 
an hyperbola when the central 
force is proportional to the dis- 
tance, 385 

Cauchy on elasticity, . . . 1 24 
solution of partial differential equa- 
tions, 201 

on differential equations, . . 214 
Centre of gravity, .' . . . .55 
its position with regard to equilib- 
rium of rotation, . . .131 
resultant moment for, . . 133 

motion of, 262 

of systems, .... 458 
of a system in a resisting medium, 474 
Central force of gravitation, . . .43 
in relation to tautochrone, . 323 
in relation to brachistoehrone, . 330 
for a point moving upon a plane, 3 78 
special cases of, which admit of in- 
tegration, . . . . .379 
forms which admit of general in- 
tegration, ..... 383 
forms which admit of integration 

by elliptic integrals, . . .389 

third form which can be solved 

by elliptic integrals, . . . 406 

Centrifugal force, ..... 245 

for brachistoehrone, . . .329 



Centrifugal force of body moving on sur- 
face, 377 

Characteristic function of motion, . .162 
its variation, . . . .166 
Chasles's shell and its attraction, . . 58 
and Gauss's theorem, . . 61 
trajectory canals, . . . .63 
analogy of attraction and propa- 
gation of heat, . . . .64 
definition of his thin shells, . 65 
potential of his shells, . . .61 
his ellipsoidal shell, ... 70 
attraction of his ellipsoidal shell, . 76 
Circle rotating with a free moving body 

upon its circumference, . . 251 
upon which a heavy body moves, 255 
rotating in a vertical position, with 
heavy body moving along its cir- 
cumference, . . . .259 
rotating with heavy body moving 

on its circumference, . . . 264 
involute, with body moving along 

it against resistance, . . . 2 74 
involute, a case of tautochrone, 325 
a tautobaryd, . . . .372 
described in a case of central 

force, 379 

horizontal, when it is in the path 
of a pendulum, . . . .419 

great, when it is nearly the path 
of a pendulum, . . . .421 

general law of description, . 432 
section of ellipsoid of inertia de- 
scribed by axis of maximum ro- 
tation area of a solid, . . . 442 
path of the gyroscope, . . 446 
Clairaut on a case of the tachytrope, . 365 
motion of a body when the central 
force is inversely as the cube of 
the distance, .... 388 

Composition of forces, .... 40 

Conclusion, ...... 476 

Condition, equations of, 24 

Cone, catenary upon, . . . .144 

tautochrone of heavy body upon, 322 
brachistoehrone of heavy body 
upon,. ..... 341 

motion of heavy body upon, . 413 



ALPHABETICAL INDEX. 



489 



Conic section described when the central 
force is proportional to the dis- 
tance, 

described when the central force 

is inversely proportional to the 

square of the distance, 

described when many central 

forces act proportionally to the 

distance 

general law of description, 
Conservation of power, .... 
' of motion of centre of gravity, . 

of areas, 

of power in motions of systems, 
Constants, variation of arbitrary, 
Continuity, solution of, in cases of resist- 
ance, ..... 
in the potential of nature, . 
Coordinates, peculiar case of, . 
Couple of rotations, .... 

of forces, 
Cusps of brachistochrone, 
Cycloid the tautochrone of a heavy body, 
the base of a cylinder on which 

lies a tautochrone, 
meridian curve of a surface of 
revolution on which lies a tau- 
tochrone, ..... 323 
the brachistochrone of a heavy 

body, 332 

a tachytrope, .... 365 
conditions of description, . 432 

Cylinder, attraction of, . . . .49 
containing a catenary, . . 143 
rotating with a body moving upon 

a given line of its surface, . 253 
having a heavy body upon its sur- 
face, 254 

vibrated by Bessell, . . . 298 
vibrated by Baily, . . 311 
containing a tautochrone upon 
its surface, . . . . 319 



385 



38G 



425 
432 
163 
242 
434 
458 
459 

273 

32 

425 

18 

40 

332 

318 

321 



D. 

Derivative multiple, . 
Determinants, theory of, 
functional, . 



196 
172 
183 

62 



Determinants applied to multiple deriva- 
tives and integrals, . . .196 

Devil on two sticks, .... 451 

Dubuat on the law of resistance of a 

medium, . . . . 292 
experiments on the pendulum 
against a resistance, . . 294 

Dupin on orthogonal surfaces, . . .79 

E. 

Economy dynamic, of nature, . . . 168 

Elasticity, 116 

Electricity, statical, . . . . .44 
Ellipse, spherical, .... 147 
described by central force which 

is proportional to distance, . 385 
described under the law of gravi- 
tation, 386 

Ellipsoid, attraction of, ... 69 

Chaslesian shell, . . . .70 
of revolution, attraction of, . 87 
of closest approximation to at- 

traction of spheroid, . . . 103 
of expansion, . . . . 118 
of reciprocal expansion, . .121 
with catenary upon its surface, 154 
with brachistochrone on its surface, 344 
defining surface of the brachisto- 
chrone, 347 

of inverse inertia, . . . 435 

of inertia, 436 

Elliptic integrals for attraction of ellipsoids, 83 
for the catenary upon the cone, . 147 
referred to spherical ellipse, . 149 
for the catenary upon the sphere, 157 
for the simple pendulum, . . 256 
for tautochrone on a moving curve, 318 
for tautochrone on a cycloidal cyl- 
inder, 321 

for brachistochrone with parallel 

forces, 333 

for brachistochrone on paraboloid, 337 
for brachistochrone on inverted 

paraboloid, .... 341 

for brachistochrone on cone, . 343 
for brachistochrone on sphere, . 346 
for circular brachistochrone, . 354 



490 



ALPHABETICAL INDEX. 



Elliptic integrals for two forms of central 

force, 389 

for third form of central force, . 406 
for motion upon a cone, . . 413 
for motion upon a paraboloid, . 416 
for motion upon an inverted para- 
boloid, 417 

for the time of spherical pendulum, 418 
for the azimuth of the spherical 

pendulum, .... 423 

for forms of force directed towards 

axis, 428 

for rotation of a free solid, . 442 

for the gyroscope, . . . 446 

for the top, .... 450 

Epicycloid a tautochrone, . . .327 

a brachistochrone, . . .331 

path described under action of 

central force, .... 379 

Equation of tendency to motion, . . 7 12 

of motion, differential, . . . 8 18 

of equilibrium, 8^ 

of orthogonal cosines, . . .15a, 

of instantaneous axis of rotation, 1 7 15 

of rotation for cylinder, . . 23 14 

of condition involved in those of 

motion and rest, . . . 26 12 

of condition referred to normal, 27 16 
of tendency to motion expressed 

by potential, .... 34 31 
of resultant, . . . . 37 21 
of potential of gravity, . . 459 
Laplace's, of potential, . . 46 3 
Laplace's, modified by Poisson, 49 2 
of potential of an infinite cylinder, 49 31 
of relation of potential to its para- 
meter, 55 4 

of Gauss, for action normal to 

surface, 60 24 

of attraction of ellipsoid in direc- 
tion of either axis, . . . 82 21 
Legendre's, for attraction of 

ellipsoid, 83i 2 

of Legendre upon attraction, . 86 10 
of function for expression of the 

attraction of an ellipsoid, . . 86 22 
of attraction of a homogeneous ob- 
late ellipsoid of revolution, . 87^ 



Equation of attraction of a homogeneous 

prolate ellipsoid of revolution, . 88 18 
of function developed in cosines 

of multiple angles, . . . 89 13 
of elementary functions of Legen- 
dre's functions, . . . 9324 
of Legendre's functions in spe- 
cial form, 99 3 

of theorem for development into 

Legendre's functions, . . 101 22 
Laplace's upon Legendre's 

functions,. . . . . 102 s 
Laplace's more general form of 

Legendre's functions, . . 102 20 
of potential of ellipsoid referred to 

centre of gravity, . . . 103 20 
of Legendre's second function, 104 3 
of external potential of spheroid 
with the introduction of ellipsoid 
of nearest attraction, . .10724 
for axes of nearest ellipsoid of at- 
traction, 108 3 

of potential for point near the 

spheroid, 110 3 

Laplace's, for spheroid which 

differs little from a sphere, . 115^ 
of ellipsoid of expansion, . . 1 1 8 8 
of surface of distorted expansion, 119 13 
of total expansion, . . . 120 27 
of ellipsoid of reciprocal expan- 
sion, ...... 121 u 

of equilibrium of translation, . 127^ 
of funicular, .... 138 14 

of catenary, .... 138 2T 

of extensible catenary, . . 141 16 
of catenary upon a surface, . 142 2 
of pressure of catenary upon a 
surface, . . . . . 142,^ 

of catenary upon a surface of 
revolution, .... 144 29 

of arc of spherical ellipse, . . 149 2a 
of total expenditure of action, . 16 2 21 
of living forces, . . . . 163 M 

Lagrange's canonical, of motion, 164^ 
Hamilton's changes of La- 
grange's canonical forms, . 165 27 
for characteristic and principal 
functions, .... 171 20 



ALPHABETICAL INDEX. 



491 



Equation of determinants, . . . 173,, 
linear solved by determinants, . 177 
simultaneous differential, related 

to linear partial differential, . 199 
differential in normal form, . 210 4 
partial differential for Jacobian 

multiplier, .... 215^ 

common differential for Jacobian 

multiplier, . . . . 21 6 2 
of Jacobian multiplier for equa- 
tions of motion, . . . 237 19 
of translation, . . . . 242 2 
of time of describing a line, . 243,,, 
of centrifugal force, . . 245 18 

of motion upon a rotating line, . 24 7^ 
of motion of a heavy body upon a 

moving line, . . . . 25 7 8 
of gain of power by motion of the 

line of support, . . . 259 25 

of motion of a fixed line through 

a resistance, .... 271 3 
of motion against friction, . 273 10 
of fixed force for tautochrone, . 31 7 9 
of tautochrone for central force, 323i 
of general brachistochrone, . 328 18 

of brachistochrone for fixed force, 328 27 
of brachistochrone for radius vec- 
tor and perpendicular from origin 
in central force, . . . 330 15 
of brachistochrone for parallel 

forces, . . . . . 331 31 
of brachistochrone on surface of 

revolution for central force, . 324 28 
of brachistochrone of given length, 34 7^ 
of brachistochrone of given expen- 
diture of action, . . . 349 10 
of the holochrone when the tune is 
a given function of the poten- 
tial, 357 14 

of tautochrone from Lagrange, 361 2 
of tachytrope, .... 364 17 
of tachytrope for central foi-ce in 

resisting medium, . . . 366 4 
of tachistotrope in resisting me- 
dium, 369 15 

of bary trope, . . . . 370 31 
of path of a point upon a surface 
with fixed forces, . . .377, 



Equation of path of a body when the force 

is central, .... 378 22 

of path of a body upon a surface 
of revolution with central force 
dh'ected toward the axis, . 4129_ 1T 
of the spherical pendulum, . 418 17 _ 21 
of force for the description of a 

given curve, .... 430*, 
of Euler for rotation of a solid, 437^ 
of living force in a rolling solid, . 45 7 2T 
of sliding motion, . . • 458 5 
of variation of arbitrary constants, 46020 
of variation of initial values of va- 
riables .... 462 lljl2 
of Hansen's method of perturba- 
tions, . . . 46 5 8 , 46635, 467 10 
of small oscillations, . . . 469 2r 
of multiplier in a resisting medium, 472 31 
of power in a resisting medium, . 474 15 
of translation of a resisting me- 
dium, 474 2J 

of rotation in a resisting medium, 475 u 
Equilibrium, equations of, . . .7 
conditions of, . . . . 29 
stable or not, . . . .30 
of translation, . , . . 127 
of rotation, . . . . .129 
oscillation about position of, . 471 

Euler, integral, 91 

note on erroneous notation, . 356 
on differential equations, . . 214 
centrifugal force on the brachisto- 
chrone, . . . v. 329 
on the brachistochrone of central 

forces, 330 

on epicycloidal brachistochrone, 331 
error regarding the brachisto- 

chrone, 353 

compound brachistochrone, . 354 
compound tautochrone, . . 358 

tachytrope of heavy body, . 364 
tachytrope for parallel forces, . 366 
tachytrope of constant velocity in 

a given direction, . . .367 

tachistotrope of heavy body, . 369 

tautobaryd of heavy body, . .373 

path of body gravitating to two 

centres, 429 



492 



ALPHABETICAL INDEX. 



Euler, equations of rotation of solid, . 437 
rotation of solid, .... 443 

E volute of the parabola a tachy trope, . 368 

Expansion, linear, . . . . .117 
ellipsoid of, . . . .118 
surface of distorted, . . . 119 
total, 120 

Expenditure of action, . . . .162 



Fontaine on tautochrone, . . .362 
Force, its origin, ..... 1 

measure of, 2 

of moving bodies, ... 4 

of nature, 28 

fixed, 28 

expressed in form, . . .29 
potential of, . . . . 29 
temporarily fixed, . . .34 

composition and resolution of, . 35 

moment of, 38 

couple of, 40 

in a plane, 42 

parallel, ..... 42 
modifying, . . . . .124 
internal, may be neglected in trans- 
lation and rotation, . . .131 
equal and parallel, in equilibrium, 132 
principle of living, . . . 163 
of perturbation, . . .459 
of moving bodies, . . .479 

central. See Central Force. 
centrifugal. See Centrifugal Force. 
Form, expressive of force, . . .29 
French, weights and measures introduced, 293 
Friction opposing motion of a body, . .270 
changing sliding to rolling motion, 458 
Functional determinant, .... 183 
Funicular, . . . . . .134 

G. 

Gamma function, . . . . .91 
note on, 356 

Gauss on action perpendicular to surface, 60 
maxima and minima of potential 

of gravitation, . . . .62 
determinants, . . . .173 



Gould's Astronomical Journal, on partial 

multipliers, 231 

on motion when force emanates 

from an axis, .... 428 

Gravitation, potential of, . . . .43 

potential for mass, ... 45 
the type of equal and parallel 

forces, 132 

its level surfaces, . . . 132 

Gravity. See Centre of Gravity. 

Gudermann on spherical pendulum, . 423 

Gyration of the devil, . . . 453 

of the hoop, 456 

Gyroscope, . . . . . . 443 

H. 

Hamilton's characteristic function, . 162 
on Lagrange's canonical forms, 164 
modification of Lagrange's ca- 
nonical forms, . . . 165 
principal function, . . .169 
new method of dynamics, . . 171 
quaternions, . . . .476 
Hansen, method of perturbations, . 465 
Helix, rotating with body moving upon it, 254 

Holochrone, 354 

Hoop, motion of, . . . . . 451 
Hyperbola, determining the limits of mo- 
tion on a rotating circumference, 265 
described by central force, . 380 
described by repulsive central 

force proportional to distance, . 385 
described by force of gravitation, 386 
Hyperboloid equilateral asymtotic, con- ' 

taining catenary, . . . 159 
defining limits of catenary upon 

other surfaces of revolution, . 160 
homofocal with ellipsoid, . . 77 
containing brachistochrone, . .347 



Inertia of matter, 1 

moment of, .... 434 

Integral multiple, 197 

of differential equations, . . 199 

Integrals, systems of, ... . 203 
elliptic. See Elliptic Integrals. 



ALPHABETICAL INDEX. 



493 



Integration of the differential equations of 

motion, 172 

Involute of circle, described in a resisting 

medium, 274 

a tautoehrone, .... 325 

Ivory on corresponding points, . .70 

Ivory sphere vibrated by Bessel, . 298 

J. 
Jacobi on Legendre's functions, . . 88 
on determinants, . . .195 
on normal forms of differential 

equations, 210 

new multiplier, . . . . 214 
principle of last multiplier, . . 228 
on the motion of a body in a resist- 
ing medium, . . . .376 
on motion of a body gravitating to 

two fixed points, . . . 429 
on motion of a system in a re- 
sisting medium, .... 474 
Jellett on the tangential radius of curva- 
ture of the brachistochrone on a 

surface, 347 

on the brachistochrone of a heavy 
body in a resisting medium, . 353 



Klingstierna's problem of the tachy- 

trope, 365 



Lagrange, method of mechanical analysis, 9 
canonical forms of equations of mo- 
tion, 165 

on determinant of derivatives, . 194 
on differential equations, . . 214 
modification of Eulerian multi- 
plier, 232 

on the tautoehrone, . . . 359 
familiar formula of the tautoehrone, 361 
on the rotation of a solid, . . 443 
on the motion of a body gravitat- 
ing to two centres, . . . 429 
on the method of perturbations 
by the variation of arbitrary con- 
stants, 459 



Lagrange on small oscillations, . . 472 
Lamina, attraction of infinite, . . .46 
Lame', relation of potential to its parameter, 55 
Laplace, equation for the potential of 

gravitation, . . . .46 

equations modified by Poisson, 48 
attraction of Newtonian shells, . 75 
functions, ..... 88 

theorems on Legendre's func- 
tions, ...... 102 

equation for nearly spherical 

spheroid, 115 

on the tautoehrone, . . . 360 

on the rotation of a solid, . . 443 

method of perturbations, . . 462 

Lead sphere, vibrated by Newton, . 293 

Legendre, attraction of Newtonian 

shells, 75 

attraction of ellipsoids, . . 83 
theorems on the attraction of ellip- 
soids, 86 

functions, . . . . .88 
special form of functions, . . 99 
Leibnitz on the force of moving bodies, 479 
Lemniscate, described under law of cen- 
tral force, 380 

Level surfaces, . . . . . 32 

of gravity, 132 

a syntachyd, . . . .375 
Limits of brachistochrone, . . . 348 
of body moving under central force, 40 7 
of heavy body on surface of revo- 
lution, 413 

Linear equations solved by determinants, 177 
partial differential equations, . 199 
equations of small oscillations, . 469 
Logarithmic spiral described by a body on 

a rotating straight line, . .251 
described against resistance, . 274 
a tautoehrone, . . . .325 
a tachytrope, . . . .365 
described under the action of a 
central force, . . . .379 

M. 

Maclaurin's attraction of ellipsoid, . 75 

Mass defined, 2 

Matter, inertia of, . . . . .1 



494 



ALPHABETICAL INDEX. 



Maupertius, action of a system, 


162 


principle of least action, 


168 


Maximum and minimum of potential, . 


29 


for equal and parallel forces, 


132 


of velocity of pendulum in a re- 




sisting medium, 


283 


Measures, French adopted, 


293 


Medium, resisting, .... 


270 


brachistochrone in, 


350 


holochrone in, , 


359 


tachytrope in, 


364 


tachistotrope in, 


369 


bary trope in, 


371 


tautobaryd in, . . . " . 


371 


synchrone in, 


374 


syntachyd in, . 


375 


systems moving in, . 


472 


Method of multipliers, .... 


25 


Hamilton's, of dynamics, . 


162 


Lagrange's, of perturbation, . 


459 


Laplace's, of perturbations, 


462 


Hansen's, of perturbations, 


465 


Modifying forces, . . . 


. 124 


Moment of force, 


38 


resultant, .... 


. 39 


of inertia, 


435 


Motion necessary to phenomena, 


1 


uniform, . r . 


2 


measure of, . 


2 


tendency to, . 


5 


equation of, . 


7 


perpetual, impossible in nature, 


31 


of translation, 


241 


of a point, : . ... 


242 


of rotation, .... 


. 433 


of a system, .... 


458 


Multiple derivatives and integrals, . 


. 196 


Multiplier, method of, . 


25 


Jacobian, .... 


. 214 


principle of last, 


228 


for equations of motion, 


. 236 


for motion of a point, 


244 


* for motion in a resisting medium, 


. 472 


N. 




Nature, forces of, . 


. 28 


Newton's shell, 


70 



Newton's experiments on pendulum, . 293 

path described when the central 

force is inversely as the cube of 

the distance, . . . . 379 

Normal form of differential equations, . 210 

Notation of reference, .... 4 

Nulty on the hoop, .... 455 

on rolling motion, . . .457 

Nutation of rotation, .... 456 



0. 



79 

30 

246 



Orthogonal surfaces, .... 

Oscillations about position of equilibrium, 
of a body on a fixed line, 
of a body on a uniformly rotating 
line, ...... 

on a rotating circumference, 

of the pendulum, .... 

of a heavy body on a rotating cir- 
cumference, .... 

of the pendulum when the resist- 
ance is proportional to the veloc- 

ity. • 

of the pendulum when the resist- 
ance is proportional to the square 
of the velocity, .... 

of the pendulum with the medium, 287 

of the pendulum when opposed by 
friction, .... 

of the pendulum observed 
Newton, .... 

of the pendulum observed 
Ddbuat, .... 

of the pendulum observed 

BORDA, .... 

of the pendulum observed 
Bessel, .... 

of the pendulum observed 
Bailey, .... 

small, theory of, 



248 
252 

256 

266 



282 



285 



290 



by 
by 
by 
by 
by 



293 



295 



296 



298 



311 
469 



P. 

Paper sphere vibrated by Dubuat, . . 295 

Parabola, path of projectile,.. . . 258 

described while rotating, . .267 

a tachytrope, .... 368 

described by law of gravitation, .379 



ALPHABETICAL INDEX. 



495 



Parabola, condition of description, . 431 
Paraboloid, brachistochrone on, . . 336 
path of heavy body on, . . 416 
Parallel and equal forces, . . .132 

Parallelopiped of translation, . . 11 

of rotation, 14 

of forces, 36 

of moments, . . . . .39 

of rotation-area, . . . 434 

Parameter of potential, . . . .54 

Perpetual motion impossible in nature, 31 

Pendulum, simple, 255 

in a resisting medium, . . 281 
seconds, of uncertain length, . 313 

spherical, 418 

spherical, related to the gyroscope, 446 
Perturbations, methods of, 459 

Planetary perturbations, case of, . 463, 465 
Platinum sphere vibrated by Borda, . 296 
Poinsot, analysis of rotation, . . 12 
relations of axis of rotation and 

of maximum rotation-area, . . 436 
velocity of rotation about axis of 

maximum area, .... 439 

on the rotation of a solid, . . 443 

Point, equilibrium of, . . . .128 

motion of, 245 

Poisson, modification of Laplace's equa- 
tion, 48 

theorem on Legendre's functions, 100 
on the pendulum in a resisting 

medium, 286 

on the top, . . . .450 

Pole of synchrone, 373 

Potential, 29 

of gravitation, . . . .45 
relation to its parameter, . . 54 

of spheroid, 99 

of equal and parallel forces, . 132 

curve, 407 

Power defined, ..... 3 

law of, 163 

gained or lost by a moving line, 259 

Pressure upon the brachistochrone, . . 329 

Principle of living forces, . . . 163 

of least action, .... 167 

of last multiplier, . . .228 

Progression, rotary, 456 



Projectile, path of, ... 410 

disturbed, 464 

Projections, theory of orthographic, . 481 
Puisieux on the tautochrone, . . .326 

Q- 

Quaternions of Hamilton promise a new 

progress to analytic mechanics, . 476 

R. 

Reference, notation of in this book, . . 4 
Residuals to express integral of central 

force, 380 

Resisting medium. See Medium. 

Resultant defined, 36 

vanishes in equilibrium of transla- 
tion, 128 

Resultant-moment, 39 

in relation to rotation, . . 130 

of gravity for centre of gravity, . 133 

Riccati on central force, . . .379 

Rolling of solid, 457 

Rotation, analysis of, . . . . 12 
combined with translation, . .16 
instantaneous axis of, . . 19 

tendency to, 40 

of expansion, . . . .120 
equilibrium of, . . . . 129 
of line upon which a body moves 
about a vertical axis, . . .261 

motion of, 433 

of a solid body, . . . " . 434 

Rotation-area, 433 

in a resisting medium, . . .475 

S. 

Screw motion includes that of all solids, 1 9 

Seconds pendulum, of uncertain length, . 313 

Sections, conic. See Conic Sections. 

Shell, attraction of spherical, . . .56 

attraction of Chaslesian, . 58 

Chaslesian ellipsoidal, . . 70 

Newtonian, .... 70 

Sleep of the top, 451 

Sliding motion, ..... 457 

Solid motion analyzed, . . . .18 

rotation of, . . . .434 



496 



ALPHABETICAL INDEX. 



Solution of a partial differential equation, 199 
of continuity in law of resistance, 273 
Sphere, attraction of, . . . .57 
having catenary upon its surface, 157 
vibrated as a pendulum, . . 294 
a synchrone, . . . 3 74 
condition of description, . . 433 
Spheroid, potential of, . . . . 99 
which is almost an ellipsoid, . .110 
almost a sphere, . . . Ill 
Spiral logarithmic path on a rotating line, 251 
logarithmic described against fric- 
tion, 274 

logarithmic a tautochrone, . 325 
a brachistochrone, . . .331 
logarithmic a tachytrope, . . 365 
logarithmic described when cen- 
tral force is inversely proportion- 
al to the cube of distance . . 379 
path of the axis of a solid, . 443 
Stability of the funicular, . . . .135 
Stader, special cases of central force,. 379 
central force inversely proportion- 
al to the cube of the distance, . 385 
central force inversely proportion- 
al to the fourth power of the dis- 
tance, 404 

central force inversely proportion- 
al to the seventh power of the 
distance, ..... 406 
Straight line, attraction of infinite, . 52 
rotating uniformly, with body mov- 
ing upon it, ... . 249 
described by heavy body, . . 255 
rotating uniformly about vertical 
axis, and described by heavy 

body, 262 

rotating uniformly about an in- 
clined axis, and described by 
heavy body, . . . .269 
a tachytrope, . . . .365 
Superposition of small oscillations, . 470 
Surfaces of the second degree homofocal, 79 
orthogonal, . . . . .79 
of distorted expansion, . . 119 
of revolution containing catenary, 143 



Sui'faces of "revolution containing tauto- 
chrone, . . . . . 322 

of revolution containing brachis- 
tochrone, ..... 334 

.with point moving upon it, . 376 
Synchrone, ...... 373 

Syntachyd, . . . .' . .375 

Systems of integrals, .... 203 

motions of, ... 458 

motions in resisting medium, . . 472 



T. 
Tachistotrope, ..... 
Tachytrope, ..... 
Tautobaryd, ..... 
Tautochrone, .... 

compound, .... 

in Lagrange's form, 

restricted by Fontaine, 
Tension of the catenary, 
Time disturbed in Hansen's method, 
Top, spinning of, . 
Translation, analysis of, . 

combined with rotation, . 

tendency to, . _ . 

equilibrium of, . 

motion of, . . 

in a resisting medium, 
Trifolia of Stader, 
Trajectory of level surfaces, . 



. 369 

364 
. 370 

316 
. 358 

359 
. 362 

139 
. 465 

449 

. -7 

16 

. 37 

127 
. 241 

474 

. 379 

32 



Variation of the characteristic function, . 166 
of a function of the elements of a 
determinant, . . . .180 

rotary, 456 

of arbitrary constants, . . 459 

Velocity, . 3 

ViEiXLE on the motion of a body along a 

rotating straight line, . . 262 
Virtual velocities, principle of, . . .7 

W. 

Weights, French, adopted, . . . 293 
Wooden sphere vibrated by Newton, 293 




Fio\ 2 




4-^ 



II a' 






-^ O 



S eS 






A • 



^ ^ 






'■\J* * -'a S> 






■'.,> j> 



.^v:,; 






**V 



S 7j 
























>0 o 












■> V 









A 






^ V* 



^ '%. 



'^ ^ 
•S*^ 



v^ *V 












tf> 






A* 



o o % 
n H -7*. 



^ 0^ 



^ o ^ 



^ ' 



■J- A \ 























>^_ 






' 


-! 


j£\ 














% 





1 

V 






"% V* 



,0 6. 



\ - N 

V 






>„ 'C- 






»:% 



***<& 



o 



x y *. 



«^ 



vV 



% 



-> A^ 



O X 



^/- ** 









<30 



3 ^ V ».,,^i0" 



v 






A' v 



•y 



V A 









^ '% 






,,r 












\ tS s -/*- 



^ 












^ ++ 



'■f 






* ^ 






%. $ 









V- 










.# . 




.\ 0c .. 


















OCT 









. 



*>. » 












%^ 






v . 



-^ 






^ v 









vV ■/' 









^ 









\ 












\V •/■ 



■:% 






J> %. 



,v> %, 









o N 
<5 +*. 















"+J. \> 






'^ ^N 






aV^,. 












■■>. ■ 






























<> * 



Va. K 



Of- <*k. 

■ 






o>* ^ 



>f> 



■**o 






^ ^ 






cv 



OCT 



- ^ 












%%*' 
















